First, you have to understand, there is no private right of action under HIPAA, so you will never have a HIPAA case, only the government can pursue the case. The government office that this responsible for this is the Office of Civil Rights in the Department of Health and Human Services. The site you can find more information at is: http://www.hhs.gov/ocr/privacy/index.html
As a result, you will be unable to pursue any damages directly from the violation.
That being said, it does appear as if the hospital might have violated your daughter's privacy rights. (Not your employer because employers are not covered entities under the law). What you should do is complain to the hospital's Privacy Officer. You can learn who this is by finding their "Notice of Privacy Practices", either on their website or by writing and asking for a copy.
Hospitals view privacy violations very seriously, and they will follow up, but the likelihood or any recovery by you doesn't really exist.
All that being stated, you might want to speak to an attorney licensed in your state to make certain that state privacy laws do not exist.
I am licensed in New Mexico and Pennsylvania, and therefore any discussion of issues related to other states must considered within that context. In addition, my comments are not intended to create a legal representation but merely to respond to the limited facts presented by the question. Any opinion herein is not meant as a precise statement of legal rights or as a recommendation of any particular course of action. A more complete legal review can be obtained through local counsel.
Your question seems to ask if you can sue, which you cannot. As my colleague notes, you can file a complaint witjh the hospital.
We do not have a client/attorney relationship until you make an appointment, we discuss your case face to face, I accept a retainer, and we explictly agree to enter into representation.
Bringing the matter to the attention of the Hospital's Privacy Officer should result in an investigation, a response to you about what's discovered and could also prevent the same thing from happening to someone else. Not a "case" - but maybe some satisfaction.
This response is intended to provide general information, but not legal advice. The response may be different if there are other or different facts than those included in the original question. See MKnutsonLaw.com for more information on why this communication is not privileged or create an attorney-client relationship.