This is not about you, please don't think it is. He is the one who has let your children down, not you. You can't keep a 'family together' when one person checks out of the relationship :(
It has been a while since you posted, so not sure what you have done in the meantime but ...
The dignified response would be to just calmly say 'I know and there is nothing you can say that will change how I feel about you now, so pack yourself a bag and leave. We can sort childcare arrangements & payments out later. There is no going back from this for us as a couple.' and do not let him see you cry or beg for him to stay.
What I would do, however, would be wait until he goes out, pack him a bag and take it around to hers ^ say what's up there and tell her she's welcome to the cheating bastard. If either of them say 'We haven't done anything/it means nothing' tell them it's not important, the fact that he's arranged it is all that matters and that he's a lying, cheating bastard and that's the end of your relationship.
There isn't any going back from this, no matter what he says, that doesn't just prolong the misery for you, make your despression worse and end like this anyway, really there isn't. Try to take it from those of us who have 'been there, done that' and completely fucked it up by 'trying again' and putting ourselves through hell - all for nothing.
Stay strong and remember this is about HIM being a fuckup, not you x