Actually, you should be turning away from her to focus on yourself and your life and your kids. As Sandi said, there is no real connection possible as long as she is pursuing another man. In order to get through this, you have to let her go, emotionally, for now...and understand that no matter what happens, that's the healthy choice for you anyway.
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How can I go to my wife and say "Why are you chatting with OM secretly?"
You don't, bro. You stop pursuing her in any way. You carry on with your life in the household and do the things to keep life going for yourself and your daughters that you need to do. And you find ways to make your life and yourself better for you and the kids.
If she wants to get close to you, if she questions you why you do not draw close to her, you might say something like, "I will not participate in an open marriage. I do not want a divorce, but as long as you're involved with another man I think it best we keep things friendly for the kids but that's about all I will do as long as you are pursuing a relationship with someone else."
