I had a discussion with my insurance agent a while ago about lending out vehicles to people (we have a van & truck with towing & trailer). What I was told was surprising - if anything happens when someone else is driving our vehicle, it goes on OUR insurance, even if that person has their own.
The damage to the vehicle, the damage to another vehicle, or property, injuries to the idiot driver we lent the vehicle to, injuries to other persons in an accident.
We were basically told flat out "don't do it". If someone needs a van or truck/trailer, do the work for them & charge them for your time/gas.
Same thing should apply here... it's nice to do favors for someone, but don't do it at the expense of your own financial security. A serious accident can easily exhaust the insurance limits, & then your own personal assets can be gone after, even if you weren't the ones driving.
My insurance agent admitted that the way vehicular insurance is set up in this country is nonsensical, but unfortunately, we can't change it by simply wishing it wasn't so.
In your situation, you & your husband should sit down & decide to either tell his ex to find her own alternative transportation, help support her (short term) with rides or busfare, or to straight out sell her the car, with transfer of title, ownership & insurance.
Because you are both married, this has serious implications on you & the children if anything were to happen while she was 'borrowing' the vehicle, & he needs to be responsible for his family now, as much as he wants to be the nice guy & help this woman out.
If necessary, call your insurance agent & verify what I have written above, & have them explain it to your husband.
Additionally, take the ex out of the picture when talking with your husband. First of all, it is clouding the picture & making him defensive against you. Secondly, it doesn't matter if she is an honor scout, fine upstanding citizen who has a perfect driving record & will assume responsibility for everything.
The insurance companies don't care, & they will go after YOU first. Even if she pays for damage, & any out of pocket expenses, your insurance will be hit, & you can have hiked rates, or even be dropped from coverage.
I'm assuming these are risks that you & your family cannot afford to take. A different means of helping his ex should be found.
Best of luck, T.