I've been in the same situation, apart from it was my own mother. Even through my pregnancies she would ask for money. I paid all of her bills including mortgage which was thankfully small at the time. She paid no bills at all, I even paid for food. She acted helpless after my dad died and put the guilt trip on.
In the end I started to give her bills back, helping her set direct debit up etc, she was very against this at the beginning but over time she's taken them all back (excluding mortgage, she rents now, long story). I paid her bills for over 10 years.
I had to put my foot down in the end, it was causing arguments with DH. Don't get me wrong she still calls me asking for money but I manage her better, and manage her expectations, but £10 here and there is much better than what I was paying which was near on £1500 a month at the time.
It will be really difficult to change her mindset, there could be multiple reasons for what she's doing, like what others have said; living beyond means, maybe even gambling who knows, I doubt she will own up to the reason.
My mother is addicted to scratch cards is what I discovered, as well as lottery. On top of that I think she was lending people money.
Your DH needs to stuck to his decision and not give in to giving money, at the end of the day he won't be appreciated any more by his mum is what I'm guessing.
Hope you get sorted, it's a stressful situation to be in.