Sorry to hear that your daughter suffer. I googled 唇系带. Some links here could be helpful to you --
http://fx120.health.china.com/Question/2006-9-17/952962.htm
http://www.doctorhe.com/xxlr1.asp?ID=7927
As to "学校就因该承担责任", I would think it will mainly depend on
1) How did your daughter fall?
2) Was it obvious that your daughter still suffering after they 简单处理了嘴唇的伤口?
3) Why did the daycare not get hold of you?
Best wishes.