Oh lovely, I'm so sorry :( I don't know you, but I can categorically say you deserve better than this.
Could you even forgive him? I know I'd end up hating my husband every time I looked at his cheating weasely face. I would become argumentive; slight grumbles about housework, money etc would include digs about how he's a cheating rat. If you did decide to stay with him, you'd need to know that you can both forgive and forget, and erase it all from your mind... otherwise it's no way to live. Personally, I couldn't and would go from a nice, placid person, to a wrathful witch.
Bear in mind that this is a man who is supposed to love you. He isn't supposed to make you feel heartbroken, unsure of yourself, and like the rug has been pulled out from underneath you. Your partner is supposed to build you up, not crush you. His devious little affair has shown no regard for you whatsoever, and he lied even when you caught him out. If he loved you - really loved and respected you - he wouldn't have hurt you like this.
Will you always wonder who he is with, if the ex is back in the area, and if he is still in contact with her? Does he love her, or will he just accidentally fall into any passing vagina now? I know I would find myself combing through text messages and emails, looking for telling body language and accusing him of all sorts, even if he was faithful from now on (and i consider myself a level headed, rational kinda gal).
I have a close family member who was cheated on; they literally went into denial as though it never happened. Still trusted their spouse with things that would make us suspicious, and entirely missed the signs two years later when another affair came to light. Even if you feel like you can still trust him, it's not worth the hurt to drag it all out, only to be cheated on again.
Finally, staying together for the kids is a terrible reason to make a go of it. Children aren't stupid; they will pick up on the vibe, the rows, the awkwardness, and they will tell you in the end that they'd rather you just ended it as opposed to dragging it out. It'll suck for them initially, but your husband should have thought about that before he decided to be unfaithful.
In the longrun, it'll work out for everyone if you put yourself first and kick him out. Besides, why spend more of your life with someone who will probably make you resent him, preventing you from moving on and finding someone fantastic down the line (maybe a Scottish, kilt-wearing Millionaire. Yum!).
I really am sorry you are in this situation. Just remember that it's not your fault, you deserve much better, and you don't have to make any decisions immediately. Take whatever time you need to regroup, but know that you have the strength to move on and be happy again xx