In 
lore it doesn't state that he can kill Rhaast.(There are only two possible outcomes: either Kayn bends the weapon to his will... or the malevolent blade consumes him completely).It say's bends it to his will which is exactly what he is doing now just bending not killing him or controlling him
I believe in both scenarios neither are destroyed completely, especially Rhaast. When Kayn wins, Rhaast is forced in a dormant state, like a coma. He loses his concious, and waits for his second taxe to consume Kayn. When Rhaast wins, Kayn loses his body's control and switches places with Rhaast in the weapon, sweeping in oblivion (where Rhaast is).
Eventually, if Kayn continues to use the darkin weapon, Rhaast will always be the winner, no matter what. When Kayn dies, Rhaast consumes the body anyway. When Shadow Assassin occurs, Rhaast still has the chance to wake up and show himself again. When Kayn loses though, it is over. His soul is not destroyed but destined in a far worse state, trapped in oblivion until end of time (or darkin weapon's magic being removed/weapon is destroyed).
It is what I think.
Well if he somehow gets rid of Rhaast he could master the shadow magics and perfect them aka becoming the Shadow Assassin we see in game
Rhaast and Shadow Assassin are two possible scenarios for the future. In-lore one of them will eventually happen, but we don't which one or when.
Edit: Nevermind. The guy below seems knows what he's talking about.
