This is the full text of the kill:

Out of the wreck rose the Black Rider, tall and threatening, towering above her. With a cry of hatred that stung the very ears like venom he let fall his mace. 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 shoulders 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 into 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 this world.

I'm not sure how you can read that and come to the conclusion that Eowyn did nothing. It's only after she stabbed him that he lost his form and the helmet fell.

It didn't kill him (you can't kill something that's already dead), it destroyed him.

ok fine, then for that matter, tell me where it says the stab to the knee destroyed him. Because the text doesn't mention that either. It just says it caused him to stumble. And later :

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 Du´nedain 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.

it just says the blade wounded him. Nothing about killing. Nothing about destroying. Your claim that the barrow blade killed him isn't supported either, yet there is ample 'circumstantial' evidence that his form was destroyed immediately after being stabbed in the face.