If NK doesn't exist in books, then there would't be a single person to kill the NK. MAybe in books, AOTD is settled differently. They have already said that there are some changes from book/show, while major points are still the same, and they won't tell us what those changes are.
Or, if Rhaegar hadn't stolen Lyanna, then Brandon would marry Catelyn Tully, and Arya would not be born. So maybe it's that.