Since the preflop raiser bet $2 into a $3 pot on the flop, then once the K hits the turn it becomes elementary to represent it with what looks like a value bet, since a paired board usually doesn't produce a player with trips shorthanded. Even AA would be hardpressed to showdown. If the opponent has the king he raises and you fold.

It doesn't matter what you have, but I would guess it's not the King for the same reason you might guess your opponent is kingless.

Think of it like this... If you have the king, and tried to check the turn to slowplay, then most of the time your opponent will check behind unless he or she is overaggressive. Against most opponents you want to value bet when you have it, and hence value bet when you don't. It has to be somehow +EV to rep it here almost every time. If the opponent smooth calls your value bet with nothing or with the king, then hammer the river regardless.

It's only my opinion that betting out like that on the turn gives you a stellar read no matter what you have.