You can hardly blame your wife for being very cautious. You betrayed her not once, but twice! How is she to really, really know that you won’t do it again? She is happy to have your help with the practical issues of life but it sounds like she hasn’t been able to bring herself to trust you with her heart. She’s therefore going out with another, and much younger, man to meet her needs for intimacy and devotion. She doesn’t believe you can do that. On the other hand, the other guy can’t possibly understand her or partner her with the children the way you can.
You don’t have to convince me of your intention to be faithful. You have to convince her. All you can do is continue what you’re doing. Be there for her. Apologize profusely and ask her what she needs you to do to have confidence in you. Then do it and give it time. If she’s willing, some sessions with a couples counselor might also help the two of you reconcile.
In the meantime, it’s wonderful that you can be involved with the children and enjoy each other as only good friends with a long history can. Only time will tell if she can forgive you enough to be your partner in every way again.
I wish you well.