KK post flop turn and river

Seat 4 is the button
Seat 1: Player1 ( $8.27 USD ) - VPIP: 16, PFR: 11, 3B: 5, AF: 2,4, Hands: 31696
Seat 2: Hero ( $5.79 USD )
Seat 4: Player4 ( $5.59 USD ) - VPIP: 20, PFR: 10, 3B: 3, AF: 4,0, Hands: 166
Seat 5: Player5 ( $5.35 USD ) - VPIP: 18, PFR: 13, 3B: 6, AF: 3,7, Hands: 372
Seat 6: Player6 ( $5.96 USD ) - VPIP: 22, PFR: 10, 3B: 4, AF: 1,6, Hands: 10834
Seat 7: Player7 ( $5.00 USD ) - VPIP: 29, PFR: 18, 3B: 1, AF: 1,9, Hands: 370
Seat 8: Player8 ( $6.04 USD ) - VPIP: 22, PFR: 13, 3B: 3, AF: 1,5, Hands: 2323
Player5 posts small blind [$0.02 USD].
Player6 posts big blind [$0.05 USD].

Should i had to bet bigger the turn? I thinked only can call 9x, full or combos draws. Ad3d, Ad5d. River better i cbeted 1/3 pot?

Poker Stars, $0.02/$0.05 No Limit Hold'em Cash, 7 Players

SB: $5.35 (107 bb)
BB: $5.96 (119.2 bb)
MP1: $5 (100 bb)
MP2: $6.04 (120.8 bb)
MP3: $8.27 (165.4 bb)
Hero (CO): $5.79 (115.8 bb)
BTN: $5.59 (111.8 bb)

Preflop: Hero is CO with K♣: K♥:
3 folds, Hero raises to $0.15, BTN calls $0.15, 2 folds

Flop: ($0.37) 9♦: 4♠: 2♦: (2 players)
Hero bets $0.24, BTN calls $0.24

Turn: ($0.85) 9♠: (2 players)
Hero bets $0.41, BTN calls $0.41

River: ($1.67) J♦: (2 players)
Hero checks, BTN bets $1.20,