Agree that what you are asking is a legal quesition not a medical question.
I would contact your state dental association (not dental board) and inquire if they have a mediation process. This is usually a better route to take since many lawyers will not take a case like this because the potential award in a claim like is this is too small for them to make enough to justify their time and efforts. You will have to bear the burden of proof that the dentist's actions were negligent and deviated from the accepted standards. As dr. Scharf stated, it is not uncommon of a file to break off in a tooth during a root canal procedure.
Personally, i would not charge for a procedure that could not be completed satisfactorily as a gesture of good will.