I think it's very strange that, since the Peter Jackson films came out, a notion has arisen that it wasn't Eowyn that destroyed the Witch-king. The theory goes that Merry's short-sword, the dagger of Westernesse that Tom Bombadil gave him, was a magic blade specifically needed to disrupt the spell that allowed to Witch-king to keep himself together. Sometimes it's claimed that the spell somehow linked the Witch-king to Sauron and this what Merry broke. In this notion, it is Merry stabbing the Witch-king that is the important moment, and Eowyn stabbing him where his face should be is not important. Anyone could have done it, once Merry had stabbed the Witch-king with his magic dagger; this is what made the Witch-king vulnerable (I shall, from here, refer to this as the 'Merry's Magic Dagger' theory).
Personally, I think Tolkien is extremely clear on this point. The main evidence for what actually happened comes from the chapter The Battle of the Pelennor Fields, in The Return of the King. Different editions of course have different page numbers, but my most-consulted edition is the 1991 3-volume set, where the important stuff is on pages 138 and 141. Your edition may differ but it's easy to find. There are two passages on the death of the Witch-king about three pages apart. For me, p.138 is where Eowyn confronts the Witch-king, p.141 is where Merry realises what has happened to the sword. In the following discussion, p.136-7 of the same chapter - where Eowyn declares "I am no man!" - is also relevant.
P.138 has the following description of the point where Eowyn and the Witch-king are facing each other:
"... Her shield was shivered in many pieces, and her arm was broken; she stumbled to her knees. He bent over her like a cloud, and his eyes glittered; he raised his mace to kill.
But suddenly he too stumbled forward with a cry of bitter pain, and his stroke went wide, driving into the ground. Merry's sword had stabbed him from behind shearing through the black mantle, and passing up beneath the hauberk had pierced the sinew behind his mighty knee.
'Eowyn! Eowyn!' cried Merry. Then tottering, struggling up, with her last strength she drove her sword between crown and mantle as the great shoulder bowed before her. The sword broke sparkling into many shards. The crown rolled away with a clang. Eowyn fell forward upon her fallen foe. But lo! the mantle and hauberk were empty. Shapeless they lay now on the ground, torn and tumbled; and a cry went up on the shuddering air, and faded to a shrill wailing, passing with the wind, a voice bodiless and thin that died, and was swallowed up, and was never heard again in that age of the world."
Then on p.141 we read this:
"...And still Meriadoc the hobbit stood there blinking through his tears, and no one spoke to him, indeed none seemed to heed him. He brushed away the tears, and stooped to pick up the green shield that Eowyn had given him, and he slung it at his back. Then he looked for his sword that he had let fall; for even as he struck his blow his arm was numbed, and now he could only use his left hand. And behold! there lay his weapon, but the blade was smoking like a dry branch that has been thrust in a fire; and as he watched it, it writhed and withered and was consumed.
So passed the sword of the Barrow-downs, work of Westernesse. But glad would he have been to know its fate who wrought it slowly long ago in the North-kingdom when the Dunedain were young, and chief among their foes was the dread realm of Angmar and its sorcerer king. No other blade, not though mightier hands had wielded it, would have dealt that foe a wound so bitter, cleaving the undead flesh, breaking the spell that knit his unseen sinews to his will."
Tolkien repeats the same words - a bitter cry, a wound so bitter; pierced the sinew, his unseen sinews; mighty knee, mightier hand - to remind the reader of the earlier description. The two descriptions resonate to remind the reader what happened a few pages before.
Merry stabbed his blade into the sinew behind the Witch-kings "mighty" knee; the Witch-king "stumbled forward" because the dagger broke "the spell that knit his unseen sinews to his will". The Witch-king stumbled; he didn't come apart, he didn't become vulnerable. He lost control of his body because he had been stabbed. It literally threw him off his stroke.
As he stumbled, he gave "a cry of bitter pain", because the sword had given him a "a wound so bitter". Not deadly or fatal or anything like; bitter - painful and unpleasant. It was superlatively bitter, for sure. Even if "mightier hands had wielded it" no other blade would have given a more bitter wound. Or as bitter a wound... which is a problem in itself. But certainly no more bitter. Not 'a wound more deadly'; not 'a wound more fateful'; 'a wound more unpleasant'.
Defenders of the 'Merry's Magic Dagger' theory claim that the barrow-blades were uniquely enchanted to harm the Witch-king, to break some protection on him or sever his connection to Sauron. But that is not what this passage says. The barrow-blades that the Hobbits got were forged by the Witch-king's enemies, yes; but so were all of the weapons (swords, daggers, axes, spears and arrow-heads) forged by the Arnorians in their 700-year war with Angmar; so were all the weapons forged by Gondor in their 1,000-year war with the Witch-king (who was, after fleeing Angmar, Lord of Minas Morgul and the slayer of the last Gondorian King). Whatever Arnorian or Gondorian weapon was used to harm the Witch-king, we can be sure the smith who made it would have been pleased. In this, the barrow-blades were no different to literally millions of other weapons forged in nearly 1,700 years of war.
As for Sauron, he has not been mentioned for at least 5 pages before "the spell that knit his unseen sinews to his will" is mentioned. It does not refer to Sauron. The Ring-wraiths are agents of Sauron, not his puppets. They may have no desires that are not his, but they have the independent capacity to carry out his instructions - "his will" means the Witch-king's will, not Sauron's.
But, the argument goes on, "no other blade" would have done as much hurt to the Witch-king.
This is, I fear, one of the places that Tolkien is just wrong. "No other blade" must include the swords of Pippin and Sam, as well as Frodo's sword that broke at the Fords of Bruinen. And yet, there is no differentiation at any point between any of these blades. We do not even know if Aragorn or the Hobbits knew which blade was which, when he returned them to Merry and Pippin at Isengard, having carried them from Parth Galen. In that passage (The Two Towers: Flotsam and Jetsam - p.209 in my edition) they are described as "the two sheathed knives" and nothing more. Not 'Pippin's sheathed knife and Merry's extra-special sheathed knife like no other'. They are exactly the same. That 'no other blade' could do such a thing makes Merry's blade unique and therefore unlike Pippin's and Sam's swords. But nowhere else is such a distinction made.
Unfortunately Tolkien does make mistakes. The number of times Gwaihir the Windlord carried Gandalf, for example - at least four, possibly five, and yet "thrice shall pay for all", apparently. It seems Tolkien forgot about the Eagle-journeys in The Hobbit when he wrote that line (The Return of the King: The Field of Cormallen, p.274).
Back to the blades... I firmly believe the line should read "No other blade, not though mightier hands had wielded it, would have dealt that foe a wound more bitter..." - not Aragorn wielding Anduril, not Eomer, not Faramir, not Theoden, not Gandalf with Glamdring, not Legolas or Gimli; and whatever sword they had used, they could not have done more than Merry with the barrow-blade. And equally, if it had been Pippin instead of Merry, the same sentence could have been written. But is this because of the blades themselves?
They are certainly in some sense 'magical' (though some of the magic of them may be because of transference). When they are found, in the chapter Fog on the Barrow-downs in The Fellowship of the Ring, we are told "... [w]hether by some virtue in these sheathes or because of the spell that lay on the mound, the blades seemed untouched by time..." (p.199).
Later, Aragorn finds the daggers at Parth Galen in the aftermath of the Orc-ambush, where they have been thrown away by the Orcs. Aragorn says that the Orcs discarded the daggers in fear, because they are "wound about with spells for the bane of Mordor" (The Two Towers: The Departure of Boromir - p.14). This sentence is often referred to by supporters of the 'Merry's Magic Dagger' theory. When it is, it is invariably misquoted. It does not say they were made with magics for the bane of the Witch-king or even of the Nazgul. It says Mordor.
The Orcs throw them away because they fear them. Why would the Orcs fear them if they were specifically magicked to harm Nazgul, or only the Witch-king? Yes, bound about with spells for the bane of unnatural evil things, the work of Mordor - Orcs and Nazgul both, we can conclude. Not bound about with spells to harm Nazgul or the Witch-king specifically, that does not make sense. Again, though, there is no differentiation between the daggers. Merry's and Pippin's have the same spells for the bane of Mordor. Merry's is not more special than Pippin's, or the other two blades that Frodo and Sam bore. They are all "wound about with spells for the bane of Mordor." Making Merry's blade unique so that "no other blade" could inflict "a wound so bitter" does not make sense in context. Indeed, in Letter 210 (written about a draft script for a Lord of the Rings film, not closely dated but placed in June 1958), Tolkien says clearly that Sam cannot stab the Witch-king at Weathertop with his barrow-blade, as if he did, the result would be the same as a reference to Bk. III; "... the Wraith would have fallen down, and the blade would have been destroyed". In other words, not 'no other blade', but precisely, 'other blades' (at least the other three barrow-blades). As a note in the same place (Note 4), Tolkien says his reference to Bk. III is to "the slaying of the Lord of the Nazgul by Eowyn". Not by Merry. Not because Merry removed any defences. Just straight up, "by Eowyn".
The point, I think, of this sentence is not "no other blade"; rather it is "not though mightier hands had wielded it". The Lord of the Rings is not, on the whole, much concerned with 'magic swords'. Anduril is a symbol of power and authority, of a history and destiny; but it is not some D&D superweapon, a +5 Sword of Orcslaying. Eomer's sword, Guthwinë, and Theoden's sword, Herugrim, are named and described but nowhere does the text suggest that they are anything more than well-made heirlooms. Eowyn's sword is not named; neither are the swords of Boromir, Faramir and Denethor, or any of the lesser heroes - Elfhelm, Erkenbrand, Gamling, Imrahil, Mablung, Damrod, Halbarad, or even Glorfindel; none are named as having 'magic swords'. Sting and Glamdring are certainly magic swords, but these are inherited from The Hobbit which is much more of an 'adventure-tale' and less a philosophical treatise on courage and sacrifice, and even then, the magic properties of Glamdring are not mentioned in The Lord of the Rings. Where Sting's magic is much used, it is remarked upon over and over. Its property of detection of Orcs benefits Sam and Frodo many times; and in the events in Cirith Ungol, the provenance of the blade in Gondolin is much remarked - the giant spiders and spider-webs of Ered Gorgoroth are specifically mentioned. The ability of Sting to cut the webs foreshadows the wounding of Shelob. No such foreshadowing exists in relation to the Nazgul. They are not noticeably afraid of the barrow-blades at Bree or Weathertop, or even Bruinen. No-one mentions any power over the undead that the blades have, at any point. Just a general 'bane of Mordor'. Anglachel/Gurthang in The Silmarillion is certainly a magic sword; but again, the mode of storytelling of The Silmarillion is very different to The Lord of the Rings. The Hobbits do not talk to their swords as Turin Turambar does, and the swords do not talk back. They do not seem to have a will of their own. Few swords are named in The Lord of the Rings, and none are described as 'magic swords' in anything like the same terms as those in The Silmarillion or The Hobbit.
Merry's deed at the Pelennor Fields is saving Eowyn's life - plucking up his courage to attack the Witch-king just at the point when Eowyn is going to die. Tolkien makes it plain; and The Lord of the Rings is a tale of courage and sacrifice, bravery in the face of evil, not about magic swords. The significance of Merry's action is in him overcoming his fear, not in fortuitously holding the only (or one of four, later three; or one of one million) 'magic dagger' that can harm the Witch-king. The providence that guided Merry and saved Eowyn was not the providence that put exactly (or approximately) the right weapon in Merry's hand, it was the fact that he was in the exact place he needed to be to chose to be brave at the right moment. Aragorn, Eomer, Eowyn, Faramir, Boromir - mightier hands than Merry's - could not have done more.
Back to the deed itself...
In Appendix A II, 'The Hourse of Eorl', the note on p.436, we read that Eowyn's shield-arm "... was broken by the mace of the Witch-king, but he was brought to nothing, and thus the words of Glorfindel long before to King Earnur were fulfilled, that the Witch-king would not fall by the hand of man. For it is said in the songs of the Mark that in this deed Eowyn had the aid of Theoden's esquire, and that he also was not a Man but a Halfling out of a far country..."The prophecy itself is contained in the Appendices. Here, the report is of the conversation between Glorfindel and Earnur (soon-to-be King of Gondor) in the closing stage of the war against Angmar. "Far off yet is his doom, and not by the hand of man will he fall", says Glorfindel (Appendix A I iii).
I think we can understand that Glorfindel saw the Witch-king's fall. He did not get a text message from Mandos or a telegram from Illuvater, there was no disembodied voice giving him information like a voicemail. We can be fairly certain that the information about the Witch-king's fall was not conveyed to him in words before he left Aman to return to Middle-earth; it must have arrived as a 'true-seeing'. He knew that at some point 'far off yet' (ie, far off in both time and space), 'someone, not a man' would accomplish the Witch-king's destruction. But because it is unreasonable to assume this knowledge arrived in verbal form, we must assume it was in the form of a vision. He saw the Witch-king's fall, and he knew it was not a man (and certainly not Earnur) who would accomplish it, because he saw who did (or, given the fact that he was seeing the future, who would) accomplish it.