In my experience in these situations they either play one of three ways:
You will give your DP an ultimatum to come clean and tell you the the truth or leave.
First scenario:
He will come clean, immediately realise what an absolute arsehole he's been and move heaven and earth to fix things. The type of conciliatory behaviour I am talking about is more than just giving you the sex life you need but things that matter. For example, he should move jobs, should change phones and for a while give you his existing phone/passwords for email etc. Anybody can make a mistake, it's how you approach fixing your cock-ups that matters. You may well decide to give him the boot anyway.
Second scenario:
He will come clean and leave you. This is a definite possibility. Whilst you cannot see it now, believe me, it is better to be on your own than living with a wanker who doesn't deserve you. It's also detrimental to the children in the long term.
Third scenario:
He will continue to bullshit you. You will continue to half heartedly believe him. Your self esteem will be shattered (if it already hasn't been so) and you will live a very unhappy life.
I am sorry if I sound really black and white about this, but I really, really do believe you deserve better.
It is totally possible to get over an affair. Unfortunately there is no possibility of you forgiving him until he has admitted what he's done and made every effort to win you back.
Your choices are difficult ones, but you know you need to make them and soon.