I don't think you should have to live with someone who is openly cheating on you.
It's offensive to me that he is openly with someone else and refuses to move out. I am offended on your behalf.
Now you need to dig deep and FIND YOUR VALUE. If you haven't seen a lawyer yet you don't know what your options are as far as forcing him out of the house, etc.
He wants two years to get the kids on a good path? He's ALREADY HAD TWENTY. If he hasn't gotten around to it by now then two more won't do him any good.
See a lawyer. A good one who will be a good backbone for you. YOU ARE WORTH IT.
Me42, H40 D12, S8, S7 A revealed: 7/13 Sep 4/14; Agreed to D 1/15