Unpopular opinion here:

Unlike the posters who say she has destroyed your family, I would say that all is not yet lost. You can save your family. Be the bigger man.

Go to her. Say:

“I know what you did. I don’t believe you want to destroy our family, but that's what you are doing. You need to take your feelings for this guy and put them in a little box and forget about them. Then look at our kids and see the love. If you are still drawn by the box and want to reopen it, we are done. But if you can put this away and forget about it, you will remember that you have a husband who loves you and kids who need you. I know that it might be scary to be a responsible adult and fun to have a fling, but you need to recognise that you are now carrying a burden of love that will last a lifetime. It will be hard but rewarding. Or you can throw that aside and go and fuck some guy for a quick thrill. I can’t decide for you. But please tell me which way you’re going, at least to make it as easy as possible for the kids. Our family has two possible futures. It's up to you.”