Q'Anilia
Exiled Master Jedi
This is my interpretation:
When you first meet the Lich King in Northrend in World of Warcraft, he says "I was once a shaman too". The quoted event in the book takes place before this encounter, so here's my guess-
When Arthas smited Ner'Zhul, he simply killed a split personality. The book states how Ner'Zhul was first Ner'Zhul, then the Lich King. This is just how Arthas was first Arthas and then the Lich King.
When then Ner'Zhul and Arthas joined together and became one (The Lich King), they attained eachothers memories. This is reason enough for Arthas to claim that he was a shaman too, since it's part of his memories and isn't directly false given how the Lich King once was a shaman.
The Lich King is the descendant of Arthas and Ner'Zhul. He's the combined being of the two, all until Arthas used the Frostmourne and destroyed Ner'Zhuls spirit.
At which point, Ner'Zhul vanishes. Arthas attain control and that's that. Two minds aren't required to control one body. Arthas doesn't need Ner'Zhul, although it was rather stupid to smite him, given how Ner'Zhul was the smart one of the two.
If Arthas really wanted power, he'd have kept Ner'Zhul around because Ner'Zhul has a much more potent mind.
Bottomline: Ner'Zhul is no more. Arthas is selfish. Arthas is the Lich King. Ner'Zhul isn't.
Ner'Zhul isn't necessarily dead though, only gone. I imagine he'll make a return which will contribute in the defeat of the Lich King.