3 Answers

Minimally, I would expect the people who run the pool to: (A) pay for the treatment of the injury; (B) immediately and obviously post it somewhere conspicuously, that no glass is to be allowed in the pool area whatsoever!

If the forementioned was the owner's response, I would honestly have to ask myself why I would want to sue someone. I'd question my motivation for seeking that level of recourse. The one thing to also take into consideration is, if legal action isn't actually needed, it tends to demand more focus and energy than people realize. Is all that worth it, if the owner responds responsibly already?

From what I read on the internet, usually sueing a place such as a swimming pool is done when there is a serious injury, such as one that cause a spinal cord or head traumas, or death.
A lawyer that is hired in such cases is a personal injury lawyer. You can look for one through the internet in your living area, ask him for more details, and find out whether sueing is practical in your daughter's case, or not.

I would see a lawyer, only after I attempted to speak with the management of the company, and seek medical bills and any followup therapy needed.  Do you have medical coverage?  If so, go through them, and let them seek payment from the pool owner company's insurance company.  That way you do not have the stress involved.