If you tell us how you found out we can come up with an alternate story to tell her.
For example, you hired a private detective, an acquaintance told you.
THIS is key------->When she tells you she has had enough........believe her.
She does not value you or your feelings. If she did, she would not, the stray cat become.
You are in a tough spot. Do not rug-sweep this. DO NOT RUG SWEEP THIS.
I don't understand the need to make her admit to it. If he knows it happened, then making her cough it up just seems like a way to humiliate her, you might as well just start the divorce.
To save the marriage, you would have to find a very safe setting, perhaps counseling, explain your commitment to the marriage, that you want to make it better, FORGIVE her, and listen very carefully to her side of it, and then bury it forever and completely. Otherwise, if you want a full admission of guilt, you're effectively establishing a narrative where your sins are acceptable and hers are not. Thus the marriage will forever be morally unequal, you will be on the moral high ground, you will always have a trump card to play if things go bad, she knows that you can tell your kids and family that she slept around, etc.; such a marriage is intolerable. No one can live the rest of their life in the doghouse for one mistake.
It shouldn't surprise you that she doesn't want to talk about. You see this as a bad sign. Actually, it's a good sign. It means she feels guilt and shame about it. That means she doesn't want to do it again.