The school holds little to no liability unless there was negligent supervision on the part of the school. The child's parents are most likely responsible for the medical bills if you can prove there was negligence and not just playing on the act of the children. If at fault, their parents pay.
I'm not real sure. You should consult a lawyer to see who will have to pay the medical bill. The lawyer can tell you more about it. I think both would have to pay since it happen at school, but it was the other kids fault. The lawyer could tell you more though.