Seriously, your boyfriend needs help. He should not have to felt like he had to sleep with his landlord. Would we really be as unsympathetic if this bloke hadn't had a colourful sexual history? Clearly sex isn't something he puts on a pedestal but it doesn't mean that prostitution is something he'd wants! And the whole "sex is the only thing I'm good at thing" sounds like he's in a really low position anyway, kind of emphasised by the fact he's depressed - your boyfriend is vulnerable and has been taken advantage of by the landlady.
You need to encourage your boyfriend to get help from his family, he needs to swallow his pride, accept a little bit of money so he can pay his rent (and perhaps get somewhere else to live in the circumstances) and start training for qualifications. If drugs are still an issue, there are several organisations that can help him. Also if he's not getting medical help for depression already.
Basically, I don't care about your relationship, as cruel as that sounds. Stay with him, leave him, do what you feel you have to, and either one is right imo. If you do leave him, you need to make sure he is getting the help he needs, even if that's just getting the support of his family. Don't just class this as cheating and leave him to get worse - yes it was wrong of him but there seems to be many layers to the issue which eventually caused him to go through with it, and he needs them resolved, not just put as part of his sexual behaviour.
Someone said this sounds like a porno, and in some ways it does. But I don't think we ever see the porn stars having such a history of problems, and huge regret and self loathing afterwards,