I don't like checking back the turn since it gives our opponent a free card with up to 6 outs (maybe more if he has the wheel draw, although that has to be discounted a bit) in a big pot where we won't really be able to fold almost regardless what comes on the river. Barring - maybe - an ace.
As played, the river must be called. As pointed out he doesn't even need to be bluffing
in order to make the call profitable; he just needs to think that his AJ is good. There are 12 combos of AJ, and only 10 combos of sets. Throw in bluffs and you've got a very profitable call.
And no, you can't really see KK coming in a situation like this. I put the likelyhood of KK/AA at about the same as 53s with the way he played it, meaning basically dismissable in the grand scheme of things, and decidedly offset by his likelyhood to bluff sometimes.