Good morning OP!
Can I first say that you sound like such an intelligent woman! You are 100% aware of what's going on and this time sounds like the straw that broke the camel's back. And thank goodness for it!
I was in a similar relationship, except mercifully, I had no children with him.
It was like a bloody revolving door here. He'd wait just long enough for me to calm down, and then he'd be back like some poor abandoned pup at the door, looking to get in again!
I think I brought up the average amount of times to leave by a considerable amount, as this surely must have happened 30 times.
Reading your OP, I just knew, or sensed that he was violent too. My instant feeling was 'oh you're in for a hammering if you let him back honey'. That you revealed that he has indeed been violent makes it all the more pressing now that you keep him gone!
Mathanxiety has given you excellent advice.
Given that it took me about 30 times to finally dump my exes sorry ass, means that despite being a little slow, I have learned a lot about what it takes.
He will try every trick in the book. He has already pulled a few - e.g. the needing medication, saying the kids hate you etc.
I would NOT ENGAGE with him at all. Try to see every message from him as 'evidence' of what a twat and twunt he is. Do not under any circumstances feel sorry for him. You know what he's like and hopefully you are prepared for what is going to come. Because, as I'm sure you know yourself, it will come. He'll beg, he'll cry, he'll apologise, he'll tell you you're mad, he'll beg some more, he'll say he just wants to be there for the children, he'll tell you that you can't afford to live without him (you can if you make sure to claim maintenance from his horrible arse), that how could you survive on your own, he'll recall one good time you had and remind you, he'll recall another time where you needed him and how good he was etc. etc. etc. On and fucking on and fucking on. So you need to take Mathanxiety's advice re setting up a new email address for communication re access to the children and leave it at that. I'm not going to repeat the advice as she has said exactly what I would say.
I should warn you (which you might already know), that he'll try to get you at vulnerable times when you're not thinking straight e.g. 3am or 6am. Once he's back in that door, it's fucking back to square one again.
What you need to do is get a claim in immediately for the 2 children who are his. How he pays it is not your problem. If he's like me ex, he'll find another fool very quickly and get his boots under her table very soon. Make sure not to get sucked in at that point either (he might send you photos of them kissing or some such other nonsense).
I rarely give the advice to contact women's aid, but I will here. They will help you on the practical and financial side of things. Just for your convenience (might as well do it now while you're thinking about it) the number is 0808 2000 247. Give them a quick call why don't you? Be warned that it can be hard to get through, so you might just sit yourself down with a cuppa and sit in for the long haul lol.
I wish you the very best of luck in this new and amazing chapter in your life. You're stronger than you know and you are certainly stronger than he knows.
Much love and strength to you. x