First, I am sorry to hear of your child's injury. I am sure that this must be traumatic for both you and your daughter.
I did not hear anything in the story that immediately says to me that you have a strong claim though. The school board likely can be found negligent and responsible under some circumstances. The exact law depends on the state. Certain states make the standard harder to meet, like requiring a showing of gross negligence, not just lack of ordinary care, to have a valid case. You will need to talk to a lawyer in your state to be sure how negligent the school employee must be for there to be a claim.
The school probably is not the cause of the children running into each other although if there was some failure to supervise to playground that might be an issue.
You sound most concerned about the nurse's failure to realize the extent of the injury. The facts sound like you did the right thing to get her seen quickly in the ER. Unless the delay caused a significantly different and worse outcome for the child then any mistake by the nurse not to get the girl x-rayed would not be the cause of any physical injury.
Although I hear you about the heartache over the situation, it is probably a case that you would have a hard time finding a good lawyer to take. That is not to say you don't have a potential claim, and you should talk to a local attorney who does injury work to be sure what your options are.
I am sorry to hear of your daughter's injury.
However, the school is not liable for children running into each other on the playground. The nurse could have done her job better, and IF the girl's injury was worsened by the delay in getting treatment, then you might have a case based on the negligence of the nurse. If, as I sincerely hope, the fractures heal up then the likelihood of recovery is not good.
Until the 1970's in this country you could not sue any governmental agency at all. The prevailing law was "sovereign immunity" based on the old idea that the king could do no wrong. Although sovereign immunity was tossed into the dust bin of history decades ago, governmental agencies still retain many defenses and immunities. Whether or not to provide additional playground supervisors out watching the children is a "discretionary function" requiring some degree of discretion and management expertise, and in such activities, governmental units still retain immunity. However, in determining whether or not to send the girl to the hospital, it can be argued that the nurse was engaged in a mere ministerial duty, and so that's why there may be some liability, even if only limited damages.
If you believe you want to go forward, then seek legal advice immediately because under governmental liability statutes, you must provide notice shortly after injuries. This is different from a statute of limitations. This is a notice requirement and can happen as quickly as 30 days after an injury depending on your jurisdiction. So, don't procrastinate, get a consultation immediately.
This answer is provided for informational purposes only. Actual legal advice can only be provided in an office consultation by an attorney licensed in your jurisdiction, with experience in the area of law in which your concern lies.