Mouse I was in your husband’s position not too long ago.
My ex-wife tried this exact same stuff after our divorce.  Her life, through her choices, became wrecked.  Constantly broke, in jail, and jumping between different rent-by-the-week motels when it came time to pay the rent.  Her boyfriend’s family (affluent, community leaders who were much better off financially than me) eventually severed all contact with them, but not before his mother tried to get me to help them out.  She (the ex) kept trying to get me to give her money, rides to ‘job interviews’, etc.  Her boyfriend called and threatened me physically if I didn’t give her money.  We had no children together, so there was no reason for me to keep in touch with her.  She was just looking for someone else to take responsibility for her life so she wouldn’t have to, and the guy she picked over me couldn’t do it.
But by that time I had a girlfriend, and she has two daughters.  We were a family, and they became the most important people in my life.  I felt bad about hanging up on my ex whenever she called, about reporting the threats to the police, and about having a lawyer inform the rich parents of the boyfriend that any attempted to contact me for money for my ex and their son would be treated as attempted extortion (I figured out that these people would give a lawyer more credence than me.)  But I felt that my new family came first, and my ex and her problems were no longer my concern.
Sleepless nights and self-recrimination about that choice?  Yes.  My ex and I were married for 13 years.  And even though she started sleeping around on me when I was on a six-month deployment (active duty US Navy,) I still felt that I could have done something to keep the marriage together.  So I had that guilt working on me too.
But there can be no reason why someone I had chosen to remove from my live could be allowed to come between me and my new family.  So I stuck to my guns, I haven’t heard from the ex in a year, and my girlfriend (now fiancee - we’re getting married in the spring) feels much more secure about our relationship.
This woman, MouseSpouse’s ex-wife, will never leave him alone as long as he caves in on any of her requests, even if it’s just a couple minutes on the phone to talk.  He must severe all contact with her, right now, or it will never end.  He’s still thinking he has a responsibility for her, and he doesn’t.  Get him to talk about it, his reasons for doing this, and don’t let him give you some half-assed one-line excuse.  I think that if he fully articulates why he feels he has to do this, he will see reason and come to this decision himself, and be strong enough to stick with it.