I was in the exact same situation as you - everything you describe. The problem with this scenario, is that he is only claiming he has an addiction because he has been caught, and he is only making promises to change for YOU, not for himself. If you and the DC were not in the picture, would he be in counselling? Would he be making every effort to stop this online activity, ON HIS OWN? I don't think so (but I guess only you can answer this).
When I first discovered my DH (now ex) doing this, I had the apologies, the promises to change, the confession that this is an addiction. He did not stop. It got worse, and he got better at hiding things. I eventually left him 4 years into our marriage, and he begged & pleaded for me to take him back. He started going to counselling and attending a sex addict group. We did reconcile...and after a while he stopped attending his group & therapy and was still hiding the problem.
Another 4 years down the road and I've left for good. Same old story "I'm sorry and I'll change," but now I don't care. A person has to change because THEY recognize they have a problem, and THEY want to get better whether you're in the picture or not. He can't do it for you, he has to do it for himself. And it's my opinion, that he needs to do it on his own, not living under the same roof as you. Sometimes a person needs to hit rock bottom before they realize how serious their problem is, if it is a problem and not just selfish and willful behaviour.
You don't need the kind of emotional damage and strain that his recovery (if he's actually serious about it) will cause you and your LO's. It will be a long, rocky road to legitimate recovery, and worse if he's just saying it to temporarily appease you. Separate yourself from him while he sorts himself out and re-examine later (but don't feel obligated to bring him back into your home even if he does get better. Do what is best for you and your children, not what you feel obligated to do).
If you are too disgusted with him to get over it, or you know that the breached trust will never be restored (and honestly it probably never will be stored because there will be relapses on the road to recovery if this is a real addiction)...it's probably best to just cut your ties with him. Trying to reconcile with my ex was a mistake for me, we ended up worse off than we were when I left him the first time.
I think it's sick that he was having a sex chat with your daughter downstairs, while he was supposed to be looking after her. Mine was on porn sites looking at images of pregnant women while I was sitting across from him, and our toddlers were playing on the carpet. It's vile.
Spare yourself and just get rid of him. You will NOT be ruining your children's lives. You'll actually be setting a wonderful example that you do not have to remain in a bad, disrespectful relationship. And as your feelings toward him have changed, what real reason do you have to stay? Staying 'for the kids' is never a good idea if things are bad.