I think it's like a crutch. When your marriage was bad, he found comfort and solace somewhere else. And now that it's better again if that is the case, he doesn't necessarily need it but why let it go? It's like a child that's grown older and no longer needs to sleep with their favorite toy or blanket but still have it in their room.
Like most people here, I'd suggest speaking with your husband about it--as difficult as it may be and having a conversation about what does he really want and then telling him what you really want. I think as a married man who isn't in a consenting open relationship he knows what he has done is a violation of your trust and his promises. And I think that your "other woman" can be one of two things: 1) duped and not aware that he isn't in an open relationship or 2) well aware that she's helping him violate your trust. But since you don't know much about her, it would be good to talk to your husband about her. Learn more about her, like you might learn about his sister (if he has one). Understand what drew him towards her and what caused him to pull away from you. This will make him feel heard and listened to and set the foundation for an on-going and open dialogue about the situation.
Before you enter this discussion, you'll most likely want to reflect deeply on what you want as well. These types of conversations never go well if you enter it emotionally. Stir the conversation away from "self-righteousness" and "anger" and "forgiveness". It should just be a discussion about how to move forward and what that looks like. Treat it like you'd have any other difficult but rational discussion about certain joint life choices such as moving or purchasing a new family car--taxes even. Look at it that way.
This will probably be a multiple step process as well and require more than one discussion. You may want to consider finding a professional mediator like a life coach or counselor to help maintain an environment that helps facilitate an open dialogue about moving forward. I don't recommend involving any family members or close familial friends either. They could transfer their own emotional state to you. This needs to be a you and your husband situation.
And perhaps in the future, you may want to sit down with "the other woman" to discuss matters with her or sit down with both your husband and the other woman. I would highly suggest doing it in a public setting and perhaps having a person outside of the situation present as well. This might also help with closure. I might also suggest you start using her name when you think about her and discuss her with your husband. She is also a human though what she has made you feel is certainly not humane. But using her actual name removes her power over you and your relationship with your husband from your perspective. The "Other Woman" title is a very powerful and rather hateful title to use which offers this mental construct you have of her to affect you thought patterns from their natural course.
But never play the comparison game. You have your great qualities and your okay qualities and your need-to-work on qualities just as your husband does and the "other woman" in this situation. And never let anyone use the term "your children". You're a mother and you know what is best for your children. Don't let them enter any conversation you have as "argument" for one option or another. They're not pawns and they deserve the happiest and healthiest home environment as possible (even if they've flown the nest).
Whatever you do, I hope you stay happy and healthy. As you said before--life has its ups and its downs but you make through. I'm certain you'll make the best dicision and come back happier and healthier and stronger because of it. Happy Holidays.