I think Jax wants to find an apprentice not only for Icathia but..
I think he wants to simply pass his skill, experience and wisdom on. He's incredibly old, and yet we don't know of any children or life partners. His backstory with Icathia is tragic but I suspect his tragedies extend beyond that...
I think he just wants a bond, an unrivaled connection to someone that he hasn't felt since Icathia and his people perished. Passing on the things he has learnt is a very intimate experience, and he wants himself to live on when (if?) he's gone.
Too much overlap in their characters to not explore this. Both had their life shattered by the Void. Both cultivated an inner strength because of it (Part of me wants Jax to be a bit world-weary so we can capture that Clint Eastwood-Hillary Swank dynamic from Million Dollar Baby). Kai'Sa thinks shes' lost a father figure and would accept another one.