Annoyed? Really?
She was not seeking death, nor was she afraid. Resigned to a certain fate, perhaps. (As she commented on, before) She realized, after Meriadoc stabbed him with a blade of the west, that she could perhaps kill him. It was a leap of faith, and a moment of certainty.
And you are commenting on the movie. The book made it clear she was explicitly seeking death. Merry even comments on it while they are riding to Gondor. When at Gondor, the Nazgul landed and tried to threaten her for being to stupid as to attack him for no living man could kill him.
She laughs in his face, removes her helmet and delivers "But no living man am I! You look upon a woman. Éowyn I am, Éomund’s daughter. You stand between me and my lord and kin. Begone, if you be not deathless! For living or dark undead, I will smite you, if you touch him."
The Witch King of Angmar is the one who is afraid even if just for a moment as he now has doubt.
The movie version made her weak and afraid while fighting him. She should have been (effectively) a viking woman, a valkyrie of types. She gloried in battle. In the movie, she only taunted him when he was about to die. Yeah, still a good line, but the entire context and importance of the line is lost.
When she awakes inside of Gondor, she is depressed because
she didn't die. (And because Aragorn chose Arwen and not her, which is why she wanted to die in the first place.)