Do we have a case if the ER doctor missed the fact that my daughter had a ruptured tendon in her hand?

Medical malpractice is negligence.  Negligence on the part of the ER doctor is the failure to exercise due care (that degree of care that a reasonable medical practitioner in the community would have exercised under the same or similar circumstances to prevent foreseeable harm).
Negligence on the part of the hospital where the ER is located is the failure to exercise due care (that degree of care that a reasonable hospital would have exercised under the same or similar circumstances to prevent foreseeable harm). If the doctor is an employee of the hospital, the employer (hospital) is liable for the negligence of its employee which occurred during the course and scope of employment.  
Prior to filing a lawsuit against the doctor and hospital, it may be possible to settle your daughter's case with the malpractice insurance carriers of the doctor and the hospital.
You should notify both of those insurance carriers in writing of your daughter's malpractice claim.
When your daughter completes her medical treatment and is released by the doctor or is declared by the doctor to be permanent and stationary, which means having reached a point in her medical treatment where no further improvement is anticipated, obtain her medical bills and medical reports (especially the medical report from the doctor who said that the "ruptured tendon was clear as day" and was "missed on the X-ray".
Your daughter's claim filed with the insurance carriers for the hospital and ER doctor should include her medical bills and medical reports  (I assume that your daughter is a minor and therefore there is no wage loss claim).
Compensation for the medical bills is straight reimbursement.  The medical reports will document the nature and extent of her injury and will be used to determine compensation for pain and suffering, which is an amount in addition to the medical bills.
If the case is settled with both insurance carriers (doctor and hospital), NO lawsuit is filed.
If you are dissatisfied with settlement offers from the insurance carriers, reject the  settlement offers and file a lawsuit for negligence against the ER doctor and hospital.  If the case has settled with one but not both parties, only name the party with whom the case has NOT settled as a defendant in the lawsuit.
You will need to be appointed guardian ad litem to file a lawsuit on behalf of your daughter if she is a minor.
If the case is NOT settled, the lawsuit for negligence on behalf of your daughter must be filed with the court prior to the expiration of the applicable statute of limitations or your daughter will lose her rights forever in the matter.