Birth Certificate problem?

My baby is a week old, today I discovered that the hospital royally screwed up and put the wrong middle name down for my daughter! It took us going to her 1 week check up to discover this. I was NEVER asked her middle name while in the hospital but my husband had written it down on the nursery card. Instead they put my first name as her middle name. I know I signed something stating all information was correct - while I was completely sick and drugged up. Why they had me even attempt to do such a thing is beyond me.
The hospital Birth Certificate is wrong (the one with her footprints), the information sent to the county for her state BC is wrong, and the information sent to Social Security is wrong! I know I can go to the court house and get the state issued BC corrected, but what about Social Security, and the fact that the hospital BC is incorrect? What can I do about those? On the hospital BC-can I white-out the wrong name and put the right name Or does that make it invalid?
Answers:
Get the state-issued birth certificate corrected, then use it to apply for a correction to her name with Social Security. The hospital one doesn't really matter.

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give the hospital a call, and let them know...see if they'd be able to at the very least, give you the name/number of where you should call...

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Whiting it out would invalidate the birth certificate.
Go to whatever County office that records birth certificates and explain to them what has happened and see if you can get it straightend out without having to go to Court (many times, it can, especially since it was only a week ago...you aren't the first person this situation has happened to and likely not to be the last either).
Most State agencies want a certified copy of a birth certificate anyway, not the certificate from the hospital (where I live, the issue a wallet sized card and it costs around $6).
As you indicate you are going to WIC on this week, take the hospital issued certificate, explain the situation and they will probably advise you to proceed with the application based on the current information (information on the BC) and indicate the issue of the wrong name on the BC and that you intend to get it corrected and indicate the new name. Once the problem is resolved, take the certified copy to them to file the appropriate forms.
The hospital certificate is not that important, what is most important is the papers are correct at the County Recorder's level.
Setting up a social security number for an infant is usually not automatic; you have to request one from the social security office, so it should not be an issue. Just wait until the matter is resolved, then file for a social security number for the child.