KK Interesting River

Hello everyone, I am curious to see what you think about this play on the turn on and the river to see if I played it optimally.

Thanks!

Known players:
[tg]Position_Stack:hands/vp$ip/RaisePreflop/AF/WTS/ORL/BBvsSteal/W$SD;SB$4.36:638/19/0/1.1/27/0/50/48;Hero$4.1:///////;MP3$4:551/12/6/2.2/26/20/100/50;BU$2.87:65/28/8/2.0/29/0/100/75;[/tg]
0.02/0.04 No-Limit Hold'em (7 handed)
Hand recorder used for this poker hand: PokerStrategy.com Elephant 0.105 by www.pokerstrategy.com.

Preflop: Hero is MP1 with K♥, K♠
Hero raises to $0.16, MP2 folds, MP3 calls $0.16, CO folds, BU calls $0.16, SB calls $0.14, BB folds.

Standard raise but I get lots of callers.

Flop: ($0.68) 9♥, 7♣, 6♣ (4 players)
SB checks, Hero bets $0.8, 2 folds, SB calls $0.80.

So here I make a large raise because the board is somewhat coordinated and there are many players in the pot. I think this play is fine.

Turn: ($2.28) J♥ (2 players)
SB checks, Hero checks.

Looking back, I should have raised again here for more protection as he will most likely pay if he is drawing (as I believe he is due to his call on the flop). I remember I checked behind to hope to induce a bluff and maintain pot control although I think it is a bad move now due to him probably drawing.

River: ($2.28) K♦ (2 players)
SB bets $1.08, Hero raises to $3.14 (All-In), SB folds.

I don't know if my shove is correct here but I think I should be safe with only a couple of hands beating me AND he checked the turn. Lots of weaker two pair and sets will pay me off. So is this hand played correctly or should I have bet the turn?

Final Pot: $6.50