Well done for asking him OP. I know that was hard for you but you can see now that, where you thought your relationship was going so well right now, he is still in denial and not ready to be completely truthful with you.
He denied it. If you had accepted that and walked away, he would have continued to keep all that secret from you. Because you pushed and pushed, he gave you a little bit more. But it won't be the whole truth. They met. What do lovers do when they meet. They don't have lunch with colleagues, that was a cover.
Anyway, you know now that he is keeping his options open. He is not fully committed to you 100% and, let's face it, why should he be when he has no consequences for his actions.
He says it is an addiction. Do you know why? Because he can abdicate responsibility, it's not his fault, he can't help himself. That is a lie and he said that because it's the only plausible lie that you might swallow.
The truth is he met her because he missed her and he wanted to be with her. He cannot tell you this because he thinks it will be too much and you will ask him to leave. He wants the best of both worlds. You at home looking after everyone and ready for more 'hysterical bonding' whenever he wants. Her waiting in the wings ready to meet when he fancies some excitement and danger in his life.
Tell him to leave. For now. Tell him you need some time and space to think about what you want. You tried reconciling and that didn't work. You have set your boundaries and he has ignored them time and time again. Tell him to give you some breathing space.
Ironically it's the best way to get him to commit to you, if that's what you want. Alternatively, with him out of the way you might realise that you don't want him back.
People who cheat are gambling. They know what they risk losing and, at the time, they decide that it's worth the risk.
It's only when their gamble fails that they are sorry. They are sorry that they got found out, not that they did it. But if there are no negative consequences he is more likely to do it again.