There is no "should" for what to do next.
It is all your choice. You have the power - take back your power.
Here are some options:
One: Make him leave, get a court order for the house, get a nursery / nanny, take him to the cleaners financially, block him everywhere and let him take you to court to see the children. This is not recommended.
Two: Make him leave, Get a court order for you to stay in the house. Let him have the children but elsewhere (his parents house?) up to 50% of the time - assuming he is willing. You can't make him step up and parent his children.
Three: Between you buy or rent a small one-bed place nearby, then take turns living in the house or the flat, i.e. the children stay in the house and the parents swap places. This can work for some couples, but it is not for everyone.
Four: Force the sale of the house, you both find new places to live. You would normally only do this if neither of you can afford to keep the house after a divorce. (A court may well insist he pays enough maintenance to keep you in the house with your children).
Five: Let him back, on the understanding that the relationship is over, he sleeps in another room, but you are both in the house to do the parenting. This is unlikely to appeal to him - he will probably only want to come back if he thinks he can continue having sex with you.
Six: Let him back, pretend to be "working to fix the relationship" whilst biding your time. Make sure he does loads of parenting. Then kick him out and divorce when you are good and ready. You need to put a cage of steel round your heart to pull this off.
A word of warning (unlikely but it is good to be careful), don't ever let him become a stay-at-home parent. He could do it for a few years, then divorce you, keep the kids, and kick you out of the home and make you pay him maintenance.