Name changed because my goodness the vipers can be cruel to those who choose to stay.
My 'D'H had an affair 13 years ago now. I poured my heart out on MN and the old relationships board crowd really laid into me for being a weak sap. It was brutal. I was really hurt and came off MN for a while to lick my wounds. However despite the way they went about it, which wasn't nice, there was some truth in what they were saying that I did gain strength from.
Understand that the fact he 'went back' means he never ended it with her. If he is seeing her every day and she hasn't dumped him it will be on going. He will have one story for you and one for her. Think along the lines of 'he is worried about your mental health so is staying with you for now, but can still see her, and he'll leave when you are more robust/the kids are are accepting/the risk of losing their jobs has receded/etc.' He will lie to you, to her, to his family, to everyone. Also understand that he will admit to the least he thinks he can get away with. So 'we just kissed' is never just kissed. My 'D'H just kissed a colleague. Or more accurately they'd been having a full affair for months and had been away on holiday several times when he was 'working' overseas. That was especially hurtful as we hadn't had a holiday in about 12 years because apparently it was too expensive.
Do you want him back? I mean really want him back on the understanding it will never be the same again. The trust might well come back if you both truly move past this but your relationship will have changed. Deep down a small part of you will always slightly despise him for being a weak man who cares only about himself. It doesn't matter how much work he does on himself you will always know that to be true. Right now you are probably in trauma mode where you desperately want everything to be as it was and are focussing on eliminating the OW without thinking outside of that. Affairs are very destructive. Trauma and PTSD are real.
If you have any chance of salvaging things you need to kick him out now, or leave. Nothings changes for him unless he gets a really big shock and loses control of the narrative. He'll either up sticks and run off with OW (in which case you have saved yourself months of trauma with him playing with your emotions while he decides whether you are 'good enough'), or he will have a horrible shock of the reality of what life post divorce looks like. No more shiny exciting affair. I bet you he'll come crawling back, but by then you may not want him. Go out, live your life, you think you love him and perhaps you do but I also think (and it was true in my case) there is an element of fear of the unknown.
Why did I stay, well I was pregnant, had given up my career to support his and financially was very trapped. Had I had a good career of my own I probably would have done things differently. We've made it work but my goodness it has been hard and it took probably 10 years to reach the point where we can genuinely forget and properly move on. 10 years!