The Schwarzschild interior solution was found not so long after the exterior solution was found. I understand that Kerr solution is significantly more complicated and there are more conditions at the boundaries but is there anything deep or profound about the interior that makes it so difficult to find a solution that describes it?
Birkhoff's theorem guarantees that any spherically symmetric interior solution you write down will have the Schwarzschild geometry as its exterior field, so all you have to do is ensure that the interior stress-energy looks reasonable for matter.
In the rotating case, a no-hair theorem guarantees that uncharged black holes have the Kerr form, but there's no result analogous to Birkhoff's theorem. Most rotating objects in fact don't have a Kerr exterior. So you either have to find one of the rare (perhaps nonexistent) objects that does, or find a vacuum solution more general than Kerr's to glue your interior solution to.
I'm not sure that this problem is actually unsolved, though. "Interior solution for the Kerr metric" by Hernandez-Pastora and Herrera (2017) seems to claim to solve it, and is published in Phys. Rev. D.