What do you want to happen, OP? In your best case scenario? Is it that he ends contact with this woman, apologises for the harm he's caused, and commits to working on improving your marriage? That's what I would want as a minimum.
You don't need proof of an affair, you don't need to know beyond a doubt that he's cheating. You have your own experience to prove to you that he is not giving you and your relationship what you need and deserve.
If I were in your situation (and I know it's easier to say it than do it, so please don't think I'm judging you, I'm not) I would tell him what I want. I would say something along the lines of, "I am really unhappy with the way things are between us. I've told you lots of times that your relationship with this other woman makes me feel disrespected and betrayed. I am not accusing you of anything. I am telling you that I cannot continue in this marriage unless you stop being in contact with her, admit to and apologise for the harm this has caused, and commit to working on our marriage. That's the bare minimum I require to keep going. If you can't do that, then I can't carry on in the marriage."
And I would mean it. Because I would not want to stay feeling so out of control and just waiting for him to decide my fate. I would take it into my own hands. You too have that power.
He may have been a good husband in the past, but he's not being a good husband now, and you deserve better than his lying, gaslighting, and cheating.
Alternatively, you stay in the marriage for your own reasons, but emotionally detach from him so that you no longer care about his cheating. I guess that's an option for some women. I wouldn't be able to do it, but I wouldn't judge someone else for staying. Just don't sacrifice your happiness and peace of mind so that he can enjoy himself with someone else.