You do not want to format the drive.
You biggest mistake, not to pick on you but rather to educate, was in thinking of the drive as a backup. A backup copy is an extra copy. Once that drive held your ONLY copy of data - it ceased being a backup. So when you know you will be putting yourself int hat situation, you make a backup first and verify it. If it happens otherwise (accident, drive failure...) your first priority is to make another backup.
Now that we have that out of the way. How important is the data?
So important that you would pay big money to not lose it then you might consider sending it out for recovery because there is always a chance that doing it yourself could cause more damage or even lose any chance of even professionals getting it back.
If you still want to attempt it yourself, do you have another drive to recover files to or one large enough to hold an image of this drive as well?
My workflow would be 1st to image the drive to a file.
Next I would run Testdisk and attempt to recover the partition.
If that fails I would use testdisk again to recover the files.
- TestDisk is like a dos program, you enter everything by commands. If you wanted something more windows like then try Recuva or photorec. But in all cases remember that you recover to another drive so as not to overwrite data on this one.
TestDisk guide:
http://www.cgsecurity.org/wiki/TestDisk_Step_By_Step (the download link is in the left pane as is PhotoRec)
Recuva :
https://www.piriform.com/recuva