Short NL 200 holdem decision, misplayed KK OOP?

[BlueBear]

Villain is VPIP%=60, PFR=16% and very aggressive postflop. Other opponents, tight aggressors. In hindsight, I probably should have raised KK a bit harder from the small blind (which I'm sure some players will do), but my read of the time is that the opponents regard me as playing tight from the blinds and that a larger raise will knock the initial raiser weak limpers out. I desired to make the pot heads-up with first raise and fortunately it happened.

The river is truly a dreadful vomit-inducing card, opponent has be completely covered and I have $170 left. Should I go to a check-folding mode as I could be behind, bet out for value, bet out to discourage a bluff or check-call to induce a bluff? I've no idea.

Seat 1: BB ( $196.15 )
Seat 2: Villain ( $371.47 )
Seat 3: MP ( $228.40 )
Seat 4: MP2 ( $191.45 )
Seat 5: Button ( $304.40 )
Seat 6: BlueBear ( $255.70 )
BlueBear posts small blind [$1].
BB posts big blind [$2].

** Dealing down cards **

Dealt to BlueBear [ Kc Kd ]

Villain raises [$4].
MP folds.
MP2 folds.
Button calls [$4].
BlueBear raises [$11].
BB folds.
Villain calls [$8].
LinkBeltBoss folds.

** Dealing Flop ** [ Js, 6d, 3s ]
BlueBear bets [$25].
Villain calls [$25].

** Dealing Turn ** [ 4s ]
BlueBear bets [$55].
Villain calls [$55].

** Dealing River ** [ Ad ]
BlueBear ???