Maybe they mean a labyrinthectomy, which is removal of the balance organ. They don't actually remove the AN itself, and is a shorter procedure, although I think it is still fairly serious and involves a stay in the hospital. There are several kinds, and they are performed specifically to address balance problems.
The loss of balance on one side takes a few weeks to get used to, but after that, the dizziness and balance problems are usually gone for good. A full surgery on the AN itself would be longer than 45 minutes, although House has done enough of them that they can probably do them pretty quickly these days.
It seems a shame to give up the good hearing to fix the balance issue, but that may be the better choice. CK does not generally affect balance much one way or the other, depending on what is happening to it. If it will clear up on its own, then CK is a gentler choice with a good chance of keeping the hearing.
Best wishes,
Steve