NL20 - River sizing?

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NL20 - River sizing?

Blinds: $0.10/$0.20 (6 Players) BB: $22.97 (Hero)
UTG: $33.02
MP: $20.00
CO: $22.07
BN: $20.00
SB: $59.97
Vilain is passive calling station pre and postflop.
Preflop ($0.30) Hero is BB with A A
3 folds, BN raises to $0.60, SB calls $0.50, Hero raises to $2.40, BN folds, SB calls $1.80
Flop ($5.40) 4 J J
SB checks, Hero checks
Betting defintively good here, I decided to Xb and pot control a bit, probably better with the As
Turn ($5.40) 4 J J K
SB checks, Hero bets $2.80, SB calls $2.80
he checks twice and my hand needs a bit of protection so I bet, we can check back but I think betting is pretty standard here.
River ($11.00) 4 J J K 5
SB checks, Hero

What's the best sizing OTR in your opinion? he can obviously have Jx or a flush but he also has a lot of Kx in his range so I think I'm missing too much value by checking (he can also call with some 4 or 5, maybe QQ...). Is it better to bet 3/4 or pretty big (something like 7/8) to extract max value of Kx

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