Having been in your position but after my exH had an affair, I would advise you to tell him to leave.
I turned myself inside out trying to understand why he did it & it was such a shock as I didn't believe he was capable of it. Our DD2 was 6 months old at the time & trying to salvage our marriage felt like the right thing to do as I was frightened, confused & didn't understand how we'd got to this point.
All my scurrying around, trying to make sense of it, fed into his sense of entitlement. He claimed to be confused when in fact, he was just a weak, self-absorbed coward - too spineless to admit that he was at fault & his coping mechanisms were the problem. He wanted to blame the affair on our marriage & rewrote history to justify it.
Now I am several months on from kicking him out, I sometimes cringe at how I was. It was what I had to do at the time but I would never do it again. If someone shows you who they are, believe them. The lack of love & respect that is shown by having an affair is terrible & if after having one, they aren't remorseful & prepared to do anything to make it right, then they should go.
I have coped. I have learnt that I am much stronger than I thought I was & the divorce ball is now rolling. I am in a strong position where I am going to get spousal maintanence until I am on my feet as my DD2 is so young. I am going to get 80% of the equity from the house as well as all my divorce costs paid my him.
The exH - in his words, his life is 'depressing', he is riddled with guilt & shame.
Get him to leave. Build your life & remember that you are worth much more than this weak, pathetic man.
Good luck.