You would most likely go nowhere by even trying to sue the family. Like I said, you would probably end up being countersued and they would probably win.
Bottom line, this girl was left unattended and was given the devices to be able to gain access to a 3 year old STALLION and was able to tack it up without anyone being the wiser.
Obviously something went wrong here, but most courts would argue that the stallion OWNER would be at fault. That would be you. The parents of the younger child (not you) could probably sue you for a hefty sum for putting their daughter in danger. It's the reality of the world.
If the mom was there, she probably saw her child in imminent danger and didn't know how to handle the situation. It's a scary thing to watch your child ride without anything going wrong. When something does go wrong, you don't see anything but it being a dangerous sport.
You said the horse spooked and ran through the fence. That is not the girl's fault. If the horse spooked, that was the horse's own action that sent him through the fence. Should the girl have had him out in the first place? No. However, that won't convince a judge to rule that the girl is at fault.
All in all, taking care of the vet bills yourself would probably be the cheapest option as well.
Please check with your state's laws regarding stallion ownership. In some states it is illegal for a minor (under eighteen) to own a stallion. Please keep that in mind because that's just one more thing that may backfire if you did decide to go to court.
I would highly suggest you look into a better situation for this horse; I.e. Having a more secure set up. Take this awful situation and learn from it.
I would suggest having latches to his stall secured somehow, or too high for a child to access. Or, as I suggested in the other thread, an alarm of some kind.
I would HIGHLY suggest that this horse be gelded as soon as possible. If he is that rank, he should be gelded. In my humble opinion, children (under eighteen) are too young to know how to handle a young, immature stallion. There is a reason for the stallion law. Please consider gelding him.