Actually Eowyn WAS poisoned. More or less like Frodo was in the first movie / book. Eowyn was found quickly and brought to the houses of healing, while Merry stumbled after them through Gondor. Pip found him ( much like in the movie ). Only the healing touch of the rightful heir to the throne could heal then. In other words Aragorn. As for why Eowyn was able to kill the witch king, it has to do with Glorfindels prophecy. The elf looked atthe Nazgul and mentioned that no living man will kill him. It's in the appendices. The Witch King references that in both book and movie.

People should ' ve read the books.