If the hard drive has really failed then there is no Linux software or any other OS that can install on it.

Knoppix does not need a hard drive to run, so you could still run Knoppix or some other Live CD, but that doesn't address your needs.

There is software that can try to recover bad sectors from drives, but even for that the drive has to be talking to the computer. If the BIOS can't find the drive I doubt that it could. And yes, trying such software under Wine seems like it would only introduce additional problems, but Wine is not the main issue here.

The important thing about backups is that they be easy, or even automatic, or they don't happen. Having an external 500 gig drive is nice, but may not be your best choice for day to day backups. I don't know anything about how you use the computer, but if you are like most users then what you consider "important data" is mostly small or reasonable size files that you have input. While it might be handy to have a back-up of your entire system on the 500 gig drive, I would consider having a small flash drive for backups of daily work (or even better, two or more that you rotate between, so that your only backup does not get erased before you discover that there is a problem). A simple script could make the back-up as easy as plugging in the small flash drive and clicking on a "make a back-up" icon. Or, if you are on a network, consider backing up key directories to a different computer. This can even be set up as an automatic process that happens each day while you are not using the computer (assuming that it is left on).