Hospital messed up. Help

When i had my baby i didnt have a valid I.d so the dad couldn't be put on the birth certificate and the baby had my last name so when I got released I had to go get an I.d. Well fast forward we were changing her last name and put dad on the birth certificate. Well about 3 days later I get a call from the hospital, turns out the lady didn't change her last name to dads last name. They said they were mailing me a $20 check to go to the health department to get it changed. We get there and they can't change it because the affidavit or whatever its called can't be changed unless through court.

What my question is, what do we do in order to change it and is there anything we can do about the hospital messing it up? Please help this has everything messed up.