In my case I knew the marriage would never be what I wanted it to be, but stayed because my h was so devastated I couldn't bear to inflict more hurt. Even though I didn't married-love him any more I still cared for him, there were the children to consider, and I'd already lived in the marriage for years without being happy anyway. There were other practical constraints too, although these wouldn't have been insurmountable if I had had the strength and guts to call it a day at that point.
So I am sorry but yes on the basis of my experience, he might just stay out of guilt and care for you, and fear for what will happen if you split (do you have children?), and confusion and exhaustion, in the hope that all will somehow work out. And yes again, staying because of guilt etc just makes the guilt grow - guilt that you are pretending a love and committment you really don't feel in your heart. However, although one reason for staying at the moment may guilt, he may also really want to stay and really still love you - it can be both, which gives you something to work on.
But I'm a bit confused - is he still seeing her? Or do you think he is being truthful about checking she is ok? It is very likely that this is the truth - remember (sorry for this, painful) he has had a relationship with her, likes her, maybe even loves her, probably misses her, feels hugely guilty.
Of course he MUST break contact with her as a basic condition of you discussing staying together, but calm-but-firm is the way to achieve this. You may need to buy a punchbag to get through the next few months.
One thing I can tell you - he probably knows how much you are suffering under all this. Knowing your pain may change his surface behaviour but it isn't going to make him love you or mend the marriage. If you constantly present him with your pain, it will get in the way of your ability to assess the marriage and fix it if it's fixable.
Ask yourself - why did he have an affair? what needs to happen so it doesn't happen again? will he do that? will you do that? you BOTH need to answer these questions to move on. For example, in Alibaldi's case it looks like her h had affairs to get some footloose and fancy free action. He needed to embrace some fundamentals of married life and parenthood - commitment and monogamy. He wasn't prepared to do this, so that was the end of the marriage. Sadly, not uncommon Alibaldi - and 100% his fault.
Be clear and calm and strong with him - he is probably desperate for some leadership and guidance. you can be a jelly here with us!