She knows you know that. Call her freaking bluff.

Articulate exactly whats happening and demand a decision.

- what a broken home does to kids

- her and the kids' stability with you vs without you

- facts about this 'grass is greener' guy: at the very least he doesn't respect her marriage now, so what makes her think he'd respect the institution if THEY get married

- ......

If you do nothing, you're marriage is over and you'll suffer. If you give her the ultimatum, there's a chance to fix things. If not, your marriage will end, but you can start healing