Selenite, I'm so sorry you are here. Truly, it's a horrible feeling and several people here, myself included, know exactly what you are going through. You have found a great place to vent, cry, ask for help, etc.
This early on post discovery, I just want to share a few pieces of advice.
1. Focus on yourself. 1 week post "D-Day" is a very hard time. You need to try to focus on yourself. You're probably not eating/drinking or sleeping. Try to eat something. A protein bar, crackers, anything. Mentally you are in a rough spot, try to physically improve yourself. Not eating/drinking or sleeping will only make your mental state much worse. Drink water regularly and try to eat something.
Sleeping is going to be much more difficult. Your thoughts will come to your head every night and anxiety/panic attacks may kick in. Try to journal during the day to get all of your thoughts our of your head. If you haven't sought after a therapist for yourself, I highly recommend you find one. There are several very good therapists that deal with this and can help you work your way through it.
2. Focus on your kids. This is a horrible event you are experiencing and extremely selfish act from your husband, but continue to focus on your children. Your kids should still be a focal point, and part of that is point 1, focusing on yourself. Be as strong as you can for yourself but also your kids.
Can you really make it work after that?
You're way to early to worry about this yet. Eventually you'll reach a point where you need to decide if you're going to stay together or divorce, but you're so early on with this ordeal.
This part may sting; what you know may not be all the details. There's a cheater methodology called "trickle truth" where once the gig is up they are only willing to give a couple of truths in order to save their "real life". You need full transparency and discover the truth. Once you have the full truth, you can make a decision about staying together or separating. But it's going to be a pain staking journey to find the truth and what you may find out during the fact finding mission may hurt. Many people here, myself included, suggest a polygraph.
He insists it was "just kissing", how can you be certain? If he's capable of doing this, can you be certain she was only one? If they are capable of being a cheater, they are capable of being a liar.
Is he still a member at that gym? If yes, why.
There's several literature out there affair spouses can read about helping their spouse cope with affairs. If he hasn't bought one yet, why?
Talk is cheap, actions mean so much more. If he's saying all the right things, that's great; is he doing the right things?
Continue to post, ask questions, etc. You've found a great support group and I'm rooting for you. But above all else, try to take care of yourself. Things will get better with time, but you need to focus on yourself.