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.
Ok let's start by putting this sentence in context. It is part of a scene where Merry watches the blade fade away after the death of the Witch-King. The actual sequence of events in the novel is quite clear:
The Witch King broke Eowyn's shield and shield-arm with his mace, then stepped forwards to deliver the finishing blow. This was when Merry stabbed him, piercing the sinews behind the knee. As his shoulders bowed from the pain of Merry's blow she struck the Witch-King between helm and mantle after which he dissipated. At the time of her blow he was showing no sign of anything more than a bitter but non-fatal wound.
Now we can go back to examine the sentence above. The wound dealt by Merry with the sword of Westernesse is described as bitter. It is not described as fatal, mortal, or killing. Tolkien was a professor of philology at Oxford, a master of language, and a contributor to the Oxford English Dictionary. The word bitter does not mean mortal, and someone with Tolkien's command of the english language would not have used it to mean mortal. If the wound were mortal he would have clearly said so: he didn't.
The next part of the sentence, "cleaving the undead flesh" is self-explanatory and I don't think needs any further comment. The last phrase is "...breaking the spell that knit his unseen sinews to his will." Merry struck the Witch-King in the back of the leg, and the blade was able to actually wound him because of its nature. The fact it was from Westernesse allowed it to work on both planes, and so it wounded the Witch-King and broke the spell that rendered him immortal. Once the spell was broken, Eowyn was able to strike the final, killing blow.
To use the terms of logic, Merry's attack with the blade of Westernesse was a necessary condition for the Witch-King's death, but not a sufficient condition. It was the combination which proved his bane, and Eowyn's was the killing blow.