The question is where do you go from here. Whatever you need from him to try and repair the damage you have to tell him.
He needs to realise what he's done, isn't just banter or excessive flirting..... It's more than that.
So I often say in these situations, ask your DH, if the situation was reversed, what would he expect and need you to do to regain trust and reassure you.
He then just needs to implement that. Now if you get a response like, "it's no big deal. It wouldn't bother me" or worse still if you get a response one cheating husband told his wife,which was if she'd been the one who cheated, he'd have divorced her and he didn't understand how she could forgive him..... Then I'd be thinking twice about it.
What your DH doesn't understand is that this behaviour of his, can backfire. The betrayed person thinks they aren't good enough and can go and seek attention of their own... Leading to an affair.
In my relationship work, I've seen it happen a fair amount.
Him taking the initiative to do things to look into his behaviour (or taking steps once directed), will certainly help. Tell him to get online and at a minimum he can Google and "how to regain trust after an affair".
Don't let him tell you they haven't done anything so it's not an affair. A good book for him to read is "Not just friends" by Dr. Shirley Glass, available on PDF.
If he values the marriage and is remorseful (rather than just sorry he was caught), then he'll do it, no question.
We all like an ego boost. Attention from the opposite sex can be flattering, but looking at where it can end up is what a reasonable person, thinking of their spouse does.
Most don't think of the consequences, because they don't expect to get caught out.
I advocate letting your spouse believe that loosing you is a possibility and one you aren't scared of, should your requirements following this affair not be met. I think if your spouse is of the belief you'd never leave, they get cocky, comfortable and they do it again.
As she's also in a relationship, you should consider letting her partner know as well. Do not tell your DH in advance, if you decide to do this. It's felt another pair of eyes kill the sneaking around and excitement from the secrecy.