Re: 44 Set in the SB
A lot of players who don't have the nuts or near nuts will slip into check-call mode if you do something that doesn't make sense, such as leading at a card that shouldn't help you. It's a tendency that makes a bet-fold line in certain situations work.
Against this particular player, it feels like her range based on the flop is KQ/QQ/KK. She doesn't sound like the sort to jam here with JT or AK. Would she with AA? Another question that you should ask yourself is if she is the sort who jams the flop with a set or slows down in order to maximize getting another bet in a bigger street.
Since her failure to raise the turn makes it look like she has KQ, I think that a possible play that you may have missed is going for the check-raise on the river.
You asked about how far you should go before slowing down. I think that The Mathematics of Poker suggested that 4-5 bets was enough with the second nuts vs a game theoretical opponent, but the exploitative strategy against a non-optimal opponent is different and you're not on the end here.
P.S. Where was this played? I think that all the places I have played allowed unlimited raising in heads-up pots only on streets that begin heads up.
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