So what do I do with a spouse who will not end it but will continue to see him or at least keep on the look out for other men?
Well, I will start by telling you what you DON'T do, Gef. You don't tell her "I will be her friend no matter what happens to us," and you don't "g(i)ve her an out tonight during our talk by telling her that if it was physical I wouldn't end it based on what I have found out about our marriage."
This is all SUPPLICATING, and it KILLSATTRACTION.
Secondly, re-read your opening paragraphs of your post. Your wife is doing all the LEADING in your relationship right now. Why are you allowing the ONE person who DOESN'T have the marriage's best interests at heart right now, to set the agenda?? Not only is that extremely damaging to your marriage, but it ALSO kills respect (and therefore, attraction) from your wife. YOU need to be leading right now, based on what you think is best for you and your girls. And what you're doing ain't it, with all due respect.
You are trying to get your wife to self-confess her infidelity. That's fine to try once, or to lay out a boundary of "I will no longer allow you to lie to our family about your affair" once you have proof. But to keep asking the cheater if they're cheating is fruitless. Because ALL CHEATERS LIE -- PERIOD. Either drop it completely, or get whatever proof you need and then confront her with a "I know all about you and ________" speech, and move on.
You need to set some boundaries, and FAST. Your wife's behavior -- openly, right in front of your daughters -- is harming them emotionally, and killing your own self-esteem. The fact that you put quotes around it, telling us of her trip to see "friends," tells us that you know EXACTLY what's going on, but yet you refuse to do anything about it.
And your girls are going to suffer. These is THE MOST IMPORTANT YEARS of their lives, developmentally, as to how they will relate to boys and men moving forward. I know you want to keep your family intact, but once you cross over into "I will do anything" territory, you've lost your moral compass and you're harming them more than helping them.
"What is best for my kids is best for me" Link to quotes: https://www.divorcebusting.com/forums/ubbthreads.php?ubb=showflat&Number=2879712