First, Welcome to SF!
I don't have much faith in this idea since there have been re-writes to the drive, but you can try any number of disk Recovery software out there.
A lot of them are free and do a decent job.
Based on what you have said, I don't think you'll have much luck, but it doesn't hurt to try- you may be able to scavenge some of it...
Googling, I got 137,000,000,hits on 'Software Recovery Free', so there are lots of things you can try...
Regarding the recovery partition that you mentioned, that will only work on the original machine. And it's not a backup- at least not in the way you would like.
Normally the OEM puts an original image of the computer on the drive on a hidden partition so in case of emergencies just like yours, the system can be brought back to its original, factory condition (with all its bloat). This will not bring back any personal data that you have accumulated. In fact, it will destroy any data that you have on the disk. (Personally, I really despise all the software that OEM's put on new computers. But, that's another story, for another time.)
It is really intended as a complete system backup (ie, factory original).
Try out some of the 'recovery' software out there and, hopefully, you'll be able to get back some of the data files you are seeking.