There is no contradiction whatsoever between the in-game lore and the novel. The novel shows the will of what was once Arthas consuming what remains of Ner'zhul and banishing Arthas's humanity. In this event, what we see is not Arthas the man taking on the incidental title of Lich King, with everything concerning Ner'zhul totally gone, but the birth of a totally new entity, the "Lich King."

Now, there is truly no "Arthas" (if by that we mean the human prince who murdered his father and merged with Ner'zhul) and no Ner'zhul (if by that we mean the Orcish shaman who was deceived by Kil'Jaeden and opened the portals that destroyed Outland); both the humanity of Arthas and the personality and will of Ner'zhul are gone, consumed by the will of Arthas to create a new, singular entity: the Lich King. What remains now are Arthas's will and memories, with Ner'zhul's memories, experience, and power subsumed into it, all combined together to form an utterly soulless creature of evil. So the Lich King can say (SPOILER) "I once sought a powerful blade," (as he does in some of the Shadowmourne whispers), and he can also say "I was once a shaman." The being that is the Lich King was once both these things, the Orc Shaman and the Lordaeron Prince; but now he is neither. Now, he is only the Lich KIng.

The scene from Rise of the Lich King only illustrates this. In the novel, we see the conflict of three wills within the mind of the newly-merged Lich King: the human Arthas, the Orc Ner'zhul, and the shattered will of Arthas, sans humanity.

In the end of the book, this conflict is resolved, with one will destroyed (Arthas's humanity, and indeed, the true Arthas, the human prince of Lordaeron), and one consumed and subsumed into the other. The will of "Arthas" is indeed triumphant in this battle, but, having consumed the will, memories, and experience of Ner'zhul, and banished everything that remained within him of the human Arthas Menethil, this being is no longer in any meaningful sense Arthas: it is now the Lich King, a new, singular being, distinct from both Arthas and Ner'zhul, and containing within it the memories and experience of both. To the extent that the will which in the end became this being was that of Arthas and not Ner'zhul, Arthas receives top billing in its creation. But its not Arthas anymore, folks. It's the Lich King, and it's got the memories and experience of both men. No contradiction there.

This, I think, is what Blizzard is going for. It's somewhat complicated, but even so, I think it shines through fairly well in the game (and also the book, properly understood).

Thank you for your time.