No experience here, but the tattoo shop should definitely compensate you for medical bills accrued due to their poor hygeine practices. Tattoo shops are very strictly regulated. For example, no food in the tattoo area, their tools should be absolutely sterilized or disposable, they should wear gloves, etc.
Also, did she get signed consent from a parent? In my state at least, signed consent is required for a minor to get a tattoo. If it's also required in your state, and the tattoo shop failed to get parental consent, that's negligence, and they're liable for that.
You're probably screwed. Most shops have you sign a waiver stating that you're of age and consenting. My advise is to hire a lawyer that takes no money up front, just a portion of the lawsuit money. If the firm thinks you have a case they will proceed. If not, there's no financial gain for them do they will tell you no.
I got MRSA during an ankle operation last April. She should be on antibiotics to kill the bacteria intravenously. I also had a wound vac. MRSA can be fatal if not treated properly. I'm still on oral antibiotics and still seeing an infectious disease Dr. and my orthopedic surgeon. It seems like a never ending process. If you know she contacted MRSA from the tattoo I would definitely sue them. I spent 3 months in the hospital and after insurance I owe $4,000.
An infection is negligence, unless she did not take care of it properly while it healed, You can sue for negligence (medical bills) even if a waiver was signed, since MRSA is not an expected consequence of getting a tatto. I assume an adult gave permission? If not, the shop is also criminally liable for giving a tattoo to a minor.