An overwhelming surplus of diggity
This is probably the best single line in Tolkein.A sword rang as it was drawn. 'Do what you will; but I will hinder it, if I may.'
EDIT: Eowyn is, it is made clear in other parts of the text, half-overwhelmed by despair. In a modern milieu, she'd have severe depression. My read, fanon if you like, is that she's so depressed already that the darkness of the mind, the fear and hopelessness, that the witch-king uses as his primary weapon, has no effect on her. She's gone through that darkness of the soul and passed through; it has no more power over here than it has already exercised.
The witch-king isn't superhumanly strong, or fast, or invulnerable. He is hopelessness and the death of hope made manifest. No man could stand against him, regardless of the strength of their arm, because they give up in the face of him. Eowyn, alone among those on the field, isn't driven by hope, but by spite. She can stab him as easily as she could stab any other sonofabitch.