OK OP you need to be clear to him (and first, to yourself).
There are two options. He commits to trying to be in a relationship with you - counselling, sex, a romantic relationship. You can mutually agree what that period of trying would look like, and how long it would go on, but it needs to have a commitment to total exclusivity and to a goal of being in a committed long-term relationship. That is what it sounds like you want.
The second option, if he decides to end the relationship, is that you split up properly, divide assets, agree custody, and live separately.
These are the only two options that will let you be sane. It sounds like he still likes you, that you 'work' together domestically and for the kids, but he does not want to be involved romantically and wants to sleep with other people. I absolutely believe he'll find someone in the short term who is ok with this set up. Then, if that's going well enough, and he can see a similar cushy domestic set up with them, he will walk away from your home. Or maybe he'll string you along for years, happy with you doing all you still do at home but unable to put any restraints on who he sees/sleeps with/what he does, because 'you're not in a relationship.'
OP I know you say its impossible to split, but you don't have a choice. What you can decide is he can't treat you in a way that will ultimately mess with your minds and those of your children. You can't model this set up to your kids, it will mess with their idea of what a functioning relationship is for years.
You need to take some time and give him a serious ultimatum for your own sake. I can totally see what he'll get out of this situation, and all I can see for you is heartache.