(((Confused))))
When you are dealing with addiction I think its a bit harder to get the WS to defog, and see what they are going to loose. With that said, I agree that you may have to leave, or even better make him leave, to start getting him to work on himself, and have a chance at R.
So to answer your question how do you get the strength?
You make a plan, you get answers, you decide what you need, and how to get it.
1. See a lawyer - Find out how a D would work, find out what your rights are, get answers to the "what if's".
2. Sit down, and figure out what exactly you need from him for him to stay at this point? Bare minimum.
You want him to delete his profiles? You want him to get STD tested? You need him to go to a therapist preferable a sex addiction therapist? Whatever. You write those things down, and you figure out the consequences if he chooses not to do those things, and if those are the things you need, then you also give him deadlines for accomplishing them. If he chooses not to even try, then he doesn't get it, and you need to throw him out. If chooses to try, but fails, then you need to stick to the consequences.
Relationships that heal, and R the WS doesn't simply get away with it. The WS takes ownership of what they did, and does the hard work of fixing themself, and with the BS the hard work of healing the relationship.
I think many FWS that have R'd can tell you they don't feel like they got away with anything, in fact many will tell you they are thankful for the BS giving them the opportunities to save themselves and their M's.
((((and strength))))