I'm not really here at the moment, but I cannot leave this unanswered. I make it big and colorful this time since this doesn't sound very safe at all.
Whatever you do, do NOT try to restore a NAND image from the faulty SD!
That could potentially be asking for serious trouble.
Save data is on the SD, not the NAND.
"SD:/Nintendo 3DS/<Long ID0 number>/<Long ID1 number/title/00040000/<title ID>/data/00000001.sav"
The ID0 and ID1 are 32 random looking hexadecimal characters, title IDs are short (8 characters).
This save files are encrypted and of no direct use – though pasting them in appropriate folders on a new SD could restore the progress. Some games using the "anti-savegame restore" anti-feature might complain and delete their progress, but that can be handled.
First thing to do is copying the content from the SD to a computer… and in case the copying does not work on any computer with multiple SD readers, you cannot get back the data yourself. Specialized data recovery software might or might not help. If the SD simply does not give any data (doesn't respond to read commands) no software method will help. The last option is expensive professional recovery (which I wouldn't recommend – no guarantee to work either).
We can't help you around that step: Without copying data to a computer: No recovery of game progress.
This thread is overly long and I feel really bad at the moment so I will not read up everything that came in the last few days.
In order to use Checkpoint, the SD has to be working in your N2DSXL. Checkpoint will then store unencrypted backups that can be restored on any CFW console and used in the Citra emulator. Using Checkpoint from time to time (followed by copying the backups to a computer) will prevent loss of game progress. When the SD went bad it is too late.
It maybe worth a try because of:
Homebrew Launcher works… so you could try the 3dsx version of Checkpoint. Maybe – and that is a big
MAYBE – it will create backups to SD:/3ds/Checkpoint and the folder can be copied to PC. But here goes another big red warning with the sign
Any usage of a damaged SD (especially writing) – no matter if hardware damaged or "just" file system corruption – carries the risk of further damaging the data – potentially beyond repair even if repair was theoretically possible before. I strongly recommend to salvage as much from SD:/Nintendo 3DS (try multiple PCs) before using the N2DSXL with that SD or even running Checkpoint to mass backup the save data. Checkpoint may crash if it can't read/write the data from/to SD 
It is your decision and your responsibility.