The trilogy is not about how the Witch King was taken down so it doesn't really make sense to focus on the fight between him and Éowyn. The fight was more extensive in the book, but when you are making a movie as huge as The Lord Of The Rings, you can't really concentrate for long on the things that are not so crucial to the plot. So the fight between the Witch King and Éowyn did not get that much time, although I'm sure it could have been awesome. On the other hand, it is kind of great that a mere woman could take the Witch King out like that.
bs the witch king can move and dosnt die when the ring is destroyed
He was dead before The Ring was destroyed, and even if Eowyn and Merry hadn't killed him, he WOULD have died when The One Ring was destroyed. The Ring was the only thing keep him, and the rest of the Nazgul, alive.