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Do you need to save it all if it is a back up ? You should still have the originals. At least it should all be nice and tidy now after all your work. Maybe best to ditch the external disk, get a new one and take a new set of backups.
And give the cat a kick to make yourself feel better
ah - so this isn't the backup - it is actually the real data for now ?
It is odd that the file names are there but nothing shows up. I would have thought the file list was generated when you plug the disk in.I would also have thought if it was physically damaged you wouldn't see anything at all.
I think you probably need to see someone who can check the disk over.
This sounds like a job for the professionals. Do you have a trusted local computer shop you can take it to? If not, I seem to remember that "OnTrack Data recovery" is one of the bigger (and probably more expensive) firms in this business. I haven't used them, so I can't offer any recommendation stronger that "I've heard of them".
Unfortunately, this is also a reminder that you should keep at least 2 backups, preferably with one off site, so you don't lose all your backups if there is a fire.