I think the real question is do you want things to work or not? If you just can't get past the idea that he had an affair and can't forgive him for that, then you should end it. If you think you will be able to get past it and he is remorseful then you two should go to counseling and try to make it work. If you have to continue living together, you should do everything you can to make it tolerable and maybe even happy if you think you can get back to that place. It's got to be having an effect on your children too, it's not like they don't notice that one of you sleeps on the couch every night. If you are going to stay together, you should definitely try and improve the situation.
Thank you! Lots of good points.