I would talk to both neighbors. Tell them about the expense to your mother and ask if they will contribute towards the expense.
I wouldn't even bring up the fact that your mother is partly at fault for letting the cat be outside (and without a leash). If they do, then don't fight their point.
As I see it, all three animals were equally outside without leashes. But that's not the point.
Two of them attacked the cat. Presumably the cat didn't initiate the attack. So the fault for the attack is with the dogs. Even if the dogs had been on leashes or in their yards, they would be to blame for attacking the cat.