Hello- Thank you for asking the question. I have over 30 years of experience working with individuals, couples and families & am happy to reply.
I am sorry to hear about this affair and the emotional turmoil that you have been enduring as a result.
I have worked with couples for over 30 years and albeit a terrible event in your life, reading what you have written is what I typically hear from women when an affair comes to light under these types of circumstances.
Denial and Blaming (you) are typical defense men used when an affair is uncovered
In all likelihood if he has been lying to you the affair is continuing however this is only conjecture based on my experience.
So how do I try to work through this with him? When he won't admit to it and stop it and move on.
If doesn't sound as though your husband is in a place of dealing with this in couples therapy so it may be best for you to find a good therapist to sort out your feelings and direction.
Why is it he stays in this marriage why not just leave?
There are many reasons and not knowing him, I cannot say for sure, but it is well know that the person with the least interest in a relationship often has the most control so that would be my sense in this situation as well.