Raise a bit larger preflop, 4.5 - 5. On the flop you're heads up and in position you want some action with top pair of KK, so a smaller bet might be better, but the BB called your bet.
What were you thinking his range was after the BB turn bet? He represents trips but what would have called your preflop raise with, A2 - there 10 of these, K2s - there are 2 of these, 22 - there only 1 of these. Unless he is real LAG player there only a small number of starting hand out of his range that give him trips. So that leave AA, KK, AK, 77 and K7 as the only other hands that beat you. I'd rule out AA and KK because he probably would have come over the top on you preflop.
You called 10.8 BB on the turn and 28.8 on the river which seems to indicate you thought you had reasonable chance of having the best hand on both cards. I would suggest raising to 25 BB on the turn. If he reraises just muck your hand. If he calls there's a good chance he'll check the river to you even with a better hand than yours. You can decide if you want to check it down or put in a small value bet (although I don't think I would).