Yeah, this should have a "Quick Format" option as well as a "No Format" option which you said you tried last time.
OK, it's time to backup your data, so you will need to remove the HDD from your PC. You could attach it to your laptop but you would need to buy an external HDD Caddy for it to work. Otherwise, look for a PC. The PC needs to be relatively the same age as your PC. A new PC is likely to have SATA connections, whereas an older PC is likely to be IDE.
Have a look at these instructions on how to remove / install a HDD:
Replacing a Hard Drive - Illustrated How to Replace a Hard Drive
When you install your HDD into another PC, the drive will appear as an additional drive in My Computer. You can then access your files from here. If you don't know where to look, let me know. Depending on what happended during the recovery you did the first time, you may find your files under a folder called "My Old Directory Structure".