AK in peculiar spot

2-4 NL table, 6 handed.

Hero = 220
SB = 120
BB = 350

Player to my left is a [censored] donkey. She's been lucking out a bunch, calling from blind with a lot of hands, betting out only when she's made something, but calling down with marginal hands. She strikes me as very passive. I've been laggish, betting into a lot flops.

Hero has AK off button, raises to 12
SB calls
BB calls
Pot: 36

Flop: Ac 4c Js

SB checks, BB checks, Hero bets 20. SB calls, BB folds. Pot = 76

Turn: 8h
Hero bets 40
BB is all in to 98
Pot 204, 58 to call.

Analysis?