I'm in a very similar situation to you, although in my case it wasn't an affair - I caught my husband on a hook up site. I know there's been no physical cheating (although I know that doesn't make it OK), and we have been trying to make it work.
Lots of people on here will tell you to leave him. And I understand why, but equally I understand it's not always that simple. All I will say is, prepare yourself because it's not going to be easy. I discovered what he'd done in October and it's basically been hell since then. The first few weeks are odd - I just felt numb and emotionless to it all and there was definitely a touch of hysterical bonding (more from his side than mine as I didn't really want him near me for a while). After the numbness wore off it was absolute hell and it's never really got easier. It's constantly on my mind and every day I question whether I should have just left him (I did throw him out at first but he begged and begged for a chance to make it work), in all honesty I partly wish I had.
Get ready for the anger as well. Some days I've been really sad and other days I'm just absolutely fucking livid and don't even want to look at him. You'll wonder what you ever did to deserve him doing this to you (the answer is nothing because the issue is all him not you, but you'll still think it anyway), you'll wonder why the hell you agreed to stay with him and out yourself through this. Because the sad fact is, I know I deserve better than this horrible feeling of unease and mistrust. You can rebuild a bit of trust (well some people do) but you will never ever trust him 100% again. That was the hardest bit for me - no matter what he does to try and make up for what he did, he can never go back and NOT do it. No matter what happens, he's always done it and you will now always know that he's capable of betraying you.
And yes he definitely needs to make changes so that he's no longer working with her - if she's leaving imminently then good but otherwise he 100% needs to find a new job. I also made it very clear to my husband that I wasn't saying 'OK this is it, we are 100% staying together' - I said I couldn't promise him that and that it needed time. You're in shock and you're going to go through a LOT of conflicting emotions, just remember you don't need to make a definite decision right now. You are perfectly within your rights to give it time and try to make it work and if it gets to a few months (or even a couple of years) down the line and you decide that actually you can't do it and you want to separate, then that is absolutely fine and your right to do so.
Please don't do the pick me dance and bend over backwards to make it work and to make him see how happy you can make him. This is on him now not you. I said all along I wouldnt do the pick me dance and I've stuck to that. If me as I am isn't good enough for him then he can fuck off. Your husband needs to be the one looking into WHY he did this, and working to make changes. I also totally understand what you mean about the emotional stuff hurting more. It's also normal to want details. If you want details, talk to him and ask for them! He needs to be talking to you about it whenever you need to (3 months down the line for me and I still need to talk about it all a LOT).
Sorry, not wanting to be negative but just being realistic about how hard this is going to be. Is he worth it? Do you really love him? If so, then I wish you all the luck in the world. Some people can and do get through stuff like this.
I realise this is getting very long now but one last thing I will add - I really strongly recommend that you get him to leave, even for a few days (but preferably a bit longer) so you can get some headspace. This is one thing I REALLY regretted not doing. My husband was only gone for a few days and looking back, I really needed more time to process things. While he was around, he was just HERE all the time. In my face so I couldnt think about things clearly. He was being really affectionate and falling over himself to do things for me and it just drove me mad.
Sorry, I'll stop now. Feel free to pm me if you ever want to chat/vent about it. I had a few people pm me when I posted my thread about it and I was so grateful, just to be able to talk to people who understood. I'm so sorry you're going through this. People who do this to their partner/spouse just have no idea how much it hurts them and turns their world upside down, I wouldn't wish the feeling on my worst enemy. Take care of yourself 