I still don't know about telling her DH, I agree he deserves to know, but the decent thing for her to do is to come clean.
Decency isn't her thing though.
If you feel certain your marriage is over, I imagine you'll not be that fussed about her H knowing , but if you were wanting to work through things ... then her H really should know.
Right now he has a wife acting inappropriately and that could fo on for a long time. Your husband is infatuated with her from the messages.
Does he generally have an obsessive personality like this? I'm not fit one moment saying she's innocent, but having that level of attention from him, must have been extremely flattering for her.
If she even had a slightly shaky marriage .... then add to that a man who seems like he worships the ground you walk on ... it would be very easy for her to cave.
Again... I'm not condoning her actions. All she needed to tell your H. Is that she's married and uninterested, so he should leave her alone.
The longer her H doesn't know, is the longer he's being betrayed.
In my experience, a confession often comes when one of the APs (affair partners), gets caught and they know the OBS (other betrayed spouse) will tell their spouse.
That gives then the opportunity to minimise the affair though...they give a watered down version.
Either way, that isn't your priority right now. You have bigger fish to fry.