Since this is a government entity, they would need to be put on notice quickly, so call a local personal injury lawyer to do so.
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I don't think that the age of the victim is what should be considered in deciding whether or not to file a lawsuit in this particular situation, but rather the injuries and any future issues that she may have with her hand as a result of said injury. Speak to a local personal injury attorney, and do not put it off (since, as my colleague as pointed out, there are very strict time restrictions when it comes to filing a claim against a governmental entity). Best of luck to you and your daughter.
If your daughter fully recovers quickly it will probably be difficult to find an attorney to take this case on a contingency fee basis as there will likely not be enough of a monetary recovery to justify the time and effort. However, if she has an extended or incomplete recovery you should pursue representation. She only has 1 year from the date of the injury to file a lawsuit to protect her rights and attorneys should be retained sufficiently in advance of this 1 year time period to adequately investigate the claim and prepare and file a lawsuit against the proper parties, assuming the matter cannot be settled. Good luck.