Do I have a medical malpractice lawsuit case

Medical or dental malpractice is negligence.  Negligence is the failure to exercise due care (that degree of care that a reasonable medical/dental practitioner in the community would have exercised under the same or similar circumstances to prevent foreseeable harm).
Prior to filing a lawsuit on behalf of your daughter for negligence against the doctor/dentist, it may be possible to settle the case with the malpractice insurance carrier.
When your daughter completes her medical/dental treatment and is released by the doctor/dentist or is declared to be permanent and stationary which means having reached a point in her treatment where no further improvement is anticipated, obtain her medical/dental bills and medical/dental reports.  I assume that your daughter is a minor and does not have a wage loss claim.  Her malpractice claim filed with the malpractice insurance carrier should include those items.
Compensation for the medical/dental bills is straight reimbursement.  The medical/dental reports will document the nature and extent of her condition and will be used to determine compensation for pain and suffering, which is an amount in addition to the medical/dental bills.  Compensation for wage loss is straight reimbursement.
If the case is settled with the malpractice insurance carrier, NO lawsuit is filed.
If your daughter is a minor and you are dissatisfied with settlement offers from the malpractice insurance carrier, reject the settlement offers and file a lawsuit for negligence against the doctor/dentist.  You will need to be appointed guardian ad litem to file a lawsuit on behalf of your daughter if she is a minor because a minor cannot file a lawsuit herself.
If the case is NOT settled with the malpractice insurance carrier, the lawsuit on behalf of your daughter for negligence against the doctor/dentist must be filed prior to the expiration of the applicable statute of limitations or your daughter will lose her rights forever in the matter.

If it broke off not due to your daughter's fault (e.g not due to her suddenly biting down on it), then there is a good chance the practice would be liable for malpractice. But malpractice cases are expensive to bring (you need to hire a medical expert to write a report and also testify). If, as we hope, the instrument can be extracted with little injury to or affect on your daughter and relatively low out-of-pocket (not paid by insurance or, as appropriate Medicaid) medical costs, it would not be worthwhile suing; only if you have large out-of-pocket costs, or some significant injury to or affect on your daughter, might it be worth suing. (If it does come to that, consult with a medical malpractice attorney about the case; many provide a free initial consultation to evaluate a case, and you can ask about that before making the appointment.)