First, don't sign any agreement and don't take back the two-month school fee from the school. If you do, you will lose any chance to sue the school and the teacher.
Second, the school and the teacher have the responsibilties to make sure all students are safe inside the shcool area and during school acitvities.
Third, get a lawyer and send a letter to the school and ask for compensation.
Fourth, make sure you stand very firm that it is not your daughter's fault. Yes, she didn't change her shoes but she is only five year olds and she needs someone to look after her all the time. Before students start any activity, teachers have the responsibility to check the students equipments and make sure they are o.k.
The school sends another teacher to talk to you because it wants to know what you will do. The school also doesn't want you to talk to your girl's teacher because it afraid the teacher may say something wrong which will increase the chance you get compensaton.
Tell school you are going to sue them. It's a private school, they would care about this law suit. It will affect their image in the community.
Find a lawyer now.
5岁女孩校内摔骨裂:学校该不该负责?