My WH has not cut contact with AP

I, 37(f), and my WH (wayward husband), 39(m), are in a complicated situation. Due to work, he is living in a different city from where we reside. He has been living there for 10 months. Initially, we tried living there with him, but some things didn’t work out, and we had to return to our city.

I want to understand if what’s happening is a "textbook" situation. On Christmas, he received a call, and that was our D-day. He started by saying it was someone looking for him, then admitted he had gone out with her and they had kissed. On the third day, he confessed they had been intimate.

In theory, he wanted to fix things. We talked, sought a therapist, and he committed to cutting contact with her. After the Christmas holidays, he returned to the city where he works. Among everything we discussed, he didn’t leave on terrible terms.

A week later, during a conversation, I asked him to show me her picture because I didn’t want to run into her without knowing who she was. He showed me her WhatsApp contact photo. In theory, there was no contact between them, and he claimed he didn’t know if she was still working there, as the year had just started.

We had a fight where he refused to give me some details about what had happened, arguing that they were private matters from their relationship. I got very upset and told him he owed her no loyalty. I even said I could ask her directly, as I had access to her phone number. I also told him that, based on her profile picture, where she was wearing her uniform, I could assume she was still working there. He kept denying it.

Later, during that same argument, I told him I’d know if she blocked me and would realize they were still in contact. In that same discussion, he admitted he had never stopped talking to her. He promised again to end it. He confessed again that they were still talking but clarified that it was only messaging and greetings at work.

It’s important to mention that he suggested sharing his constant location so I’d know where he was, assuring me he’d always answer my calls. He also installed a camera in his living room, showing the apartment door, so I could be sure she hadn’t been to his apartment again.

I don’t understand anything. My WH says he doesn’t know what to do and seems unable to decide what life to choose. He says he still wants to do things and have someone to do them with, and that I don’t seem to want the same. It’s worth mentioning we have two kids (5 and 8 years old). I’ve told him it’s impossible for us to have that life he has over there—free of responsibilities, going out every day, drinking whenever he wants. That’s not our life. It’s not his life. But for now, he’ll be working in that city until August.

I want us to reconcile, rebuild our marriage, and fix what needs fixing. Even though I believe the infidelity is his responsibility, we’ve talked, and I’ve noticed things he wished had been different. But he doesn’t seem to be at that point. He’s still in contact with the AP.

I would like to hear everyone’s thoughts. I don’t know if I’m doing the right thing by insisting on reconciliation or if this is a lost cause and he’s just waiting for me to end it. For the record, I’ve already asked him that, and he says he’s not staying in this situation with that purpose.

I really don’t know what to do. I’m looking for advice.