Only way I know is the Replace the mechanical SD Card Holder. This requires soldering and should only be preformed by a qualified technician.

You can look up the parts on E-Bay thou you still need someone to un-solder the old part and re-solder the replacement.
Local technicians will not do this kind of task with your own part, as they make money on the markup of the part itself, and their prices are outrageous.

However you can take it to a Tech School and ask the teacher, if you could donate a small amount for one of the students to perform this task.

You may even know a electronics qualified individual it may even be you.

I am a technician and know that if you try to repair the existing SD Card Holder it still has to be removed from the circuit, an after repair most likely will not function correctly anyway.

But in my opinion from what you said it is broken and needs replaced.
Start a search by entering "Molex SD Card surface mount holder". Now remember they have many types SDHC and MicroSD, Push-in Push-out ect....

You should be able the tell by looking to find what you need. There is about a dozen solder connections the size of a cat's whisker.