Perhaps. It depends on the facts and circumstances of her fall. You should contact a local personal injury attorney to discuss the school's potential liability. Keep in mind that if this is a public school, there are very short notice of claim requirements which, if not complied with, may prevent you from pursuing a claim.
Although there is no question, it can be assumed you are inquiring as to whether or not you have a viable claim. You should consult with a personal injury attorney as soon as possible as there may be short filing times involved. I'm sure there is a lot to this story which really requires a face to face consultation.
If you are the owner or an owner, under New York law you could be passively liable for the injuries or damages caused by your friend. Make sure you report any accidents to the insurance company covering the vehicle at the time of the accident. If timely reported, you'll be covered and they'll deal with the claim.
Is this a case you're handling yourself? If you are, you shouldn't be. If you have an attorney, he or she should have put the manufacturer on notice and investigated the possibility of successfully pursuing a products claim. Make sure the sink itself has been preserved and is available to determine why it failed. If your attorney is not looking at this aspect of a claim, perhaps you should consider a second opinion.
You are not "charged" in the criminal sense but you are liable in money damages. If you receive a letter or legal process concerning a particular incident, turn it over to your insurance company immediately. If there was an accident, make sure your carrier is aware of it. Not telling them in a timely fashion may result in the insurance company denying coverage in which case you may become personally liable.
Make sure she gets the medical attention she needs and all relevant insurance companies are put on notice. The statute of limitations for her is tolled until she reaches the age of majority. If something accident related develops in that time, you can always proceed with a claim. You do not indicate if the vehicle was private or public. If the latter, her age will not toll the time to file a notice of claim in which case you should seek the services of an attorney as soon as possible.
The fact that you fell and sustained injury does not, by itself, determine whether or not you have a case. The lawyer may be investigating to figure out who, if anyone, may be liable for your fall. Also, since this occurred such a short time ago, your injuries may not have fully developed. You are always free to change lawyers but a month is not a long time for a lawyer to do a proper investigation. Contact him or her and then, if you're not satisfied, consider asking someone else to take a...
In NY, your exclusive remedy for on the job injuries is workers compensation unless you can show your employer intended to cause your injury or provided no comp coverage which is obviously not the case. As indicated by my colleague, you may have a claim outside of worker's comp if you can show someone other than your employer caused your injury.