the only thing even remotely worth a try would be an excellent data recovery program called Recuva (by Piriform).
but no program is going to read a physically damaged drive - and your drive is VERY physically damaged.
all recovery software that I've used relies on the drive having a drive letter assigned to it by the system.
so first up you would need to initialise AND format that drive. both of those processes would be SEVERELY hindered due to the damage (which is how it lost them in the first place).
and the mere fact that you are accessing the drive will further impact data recovery chances.
then, let's say all the rainbow coloured unicorns formed up in a perfect row, you are then up for hours and hours of time with Recuva going through the sectors looking for files.
and it won't find fileA and fileB - these sort of programs find fragments of files and have no way of knowing how to 'stitch' them back together, so forget pictures, music and video files, they'll all be 'corrupted'.
so even if you could get the PC to detect the drive, assign a drive letter, and run recovery software, you will after all that - at best - have a bunch of corrupted files you can't use.
and a paid-for professional recovery mob will charge 100's of dollars - so is the data worth that?
this is a classic sordid tale of a situation totally avoidable by having a backup regime in place - but that doesn't help you know.