Poker ACR $10 on demand spot with KK

WPN, 75/150 blinds, 15 ante No Limit Hold'em Tournament, 9 Players
Poker Tools Powered By Holdem Manager - The Ultimate Poker Software Suite.

SB: 9,832 (65.5 bb) - 30/29 - playing really loose passive.
BB: 1,570 (10.5 bb)
UTG+1: 2,311 (15.4 bb)
UTG+2: 285 (1.9 bb)
Hero (MP1): 7,290 (48.6 bb)
MP2: 975 (6.5 bb)
MP3: 1,100 (7.3 bb)
CO: 7,005 (46.7 bb)
BTN: 10,099 (67.3 bb)

Preflop: Hero is MP1 with Kc Kh
2 folds, Hero raises to 375, 4 folds, SB calls 300, BB calls 225

Flop: (1,260) 6h Ts 5d (3 players)
SB checks, BB checks, Hero bets 625, SB calls 625, BB folds

Turn: (2,510) 2h (2 players)
SB checks, Hero bets 1,999, SB calls 1,999

River: (6,508) 4c (2 players)
SB checks, Hero checks

Results: 6,508 pot
Final Board: 6h Ts 5d 2h 4c
SB mucked and lost (-3,014 net)
Hero showed Kc Kh and won 6,508 (3,494 net)

Did i miss a value bet on the river or is a check to showdown ok.
Does SB call with much worse ? if so how much should i be betting on the river.
Is my sizing on the other streets ok or should i make it bigger for a river shove ?

Source.