Loualea, I really feel for you. There is no easy answer. Based on the situation you are describing, I cannot see what would be gained in exposing the affair.
Exposure would make reconciliation very difficult if not impossible, and that is clearly what you want. Don't do it!
I do, however, wish I communicated my standards to my W more directly and clearer. I thought DBing was about empathy and understanding you're spouse. I thought it was about forgiveness. It is about these things, but in my case, I confused them with tolerance. I see absolutely no reason for anyone to tolerate, or ignore an affair.
When you're H asks "When should I come home". The answer should be straight forward and direct:
"Not until the affair ends and you are serious about repairing this marriage. I will not tolerate having a relationship with someone involved in an affair"
I know how thought this can be. You don't want to believe the person you love is treating you this way, or would honestly choose a fragile new shallow relationship over you're deep mature relationship. You hope that they will see how understanding you are being and reciprocate that understanding.
This is not the reality. Right now they are thinking only of themselves. They are convinced that there is no future with you. They see no value in you as a person. They are mentally doing everything they can to justify their behavior by devaluing you. This is the reality.
You cannot have a relationship with someone in this frame of mind. You need to establish firm clear boundary's, communicate them and enforce them.
If I had done this up front, my marriage probably would still have failed. But, it would have moved in that direction sooner, and I would have saved myself a tremendous amount of heartache.
Again, I feel for you. And regardless of what you do, don't be too hard on yourself. You are not driving this bus.