I think that moving on as if it's over with is a bad idea. It's far from finished until he chooses to actually cut off ties with her related to anything outside of what would be necessary (assuming she still sings for him?) I will be honest, it helped when you clarified that he is trying to show disinterest in her texts but she's persistent, adn that's a problem. I feel if I'm giving him the benefit of the doubt, that he's trying, without burning the bridge with her as a friend. When I say giving him the benefit of the doubt, I mean that this would be assuming he was being honest when it comes to the idea that she pursues. This may be the case as there are women like this. Well, I'm not going to single out women because it's not exclusive to the gender. But anyway, here's my take from this point.

Your husband could have made the mistake of "enjoying" the playful attention from her, since he figured he wasn't "doing" anything (which is not true it's him being in denial really) but anyway since he figured he was not pursuing, he just let it be and enjoyed the attention. It's a temptation for a lot of people to have this kind of flirtatious person paying attention to us. Still wrong but that's established. I think he got caught in the situation and since you didn't know it was easy to ignore until such time he was caught.

You cannot just let this go. That would be making the entire ordeal, and his behavior minor and inviting further transgressions later. You need to make it clear right NOW and let him know this is not to be tolerated and although having female friends is ok (clarify this), female friends that pursue like this and flirt hard with him are not. You really should demand he make it clear to her very overtly that her behavior and pursuit of him needs to stop and he needs to, show his devotion to you adn loyalty by complying.