Jailer would fall to the same fate as anyone else would with the one ring.
He would seek to use it's power to remake the world as he originally intended, but the second he accepts the ring it's over, the darkness will bind him.
The only way to successfully beat Sauron is to destroy the ring and resist it's corruption. None of the characters both Lich King and Jailer seem capable of doing that.
The only ones who I can see tossing the ring out are Jaina and Thrall. That's it.