I recently had to fix someone’s external hard drive. I checked Google on the model and found a lot of posts about problems which all boiled down to failing power supplies. I replaced the power supply, and it started working again. However, it did make strange noises when it was not working correctly, so it may not be the problem in your case. Google is very useful for checking out these kinds of problem.
Otherwise, I have done a lot of data recovery for people using free tools off the Internet such as such as Recover My Files or EasyRecovery (I have a boot CD with these kinds of tools on it). Unless the drive is physically damaged (in which case it often makes weird clicking noises), you can usually get all the data off of it with some sort of recovery tools by searching the drive in “raw” mode and copying everything it finds to another drive. I have seen drives mysteriously lose their MFT (Master File Table) or partition table and this is the only way to get the data back when this happens. The good news is that it usually works and doesn’t cost you anything except time to fix. It even works if you formatted the hard drive.
Those are my words of wisdom on the subject and I hope it helps you out.
I would run chkdsk on the drive, that should help with rebuilding the indexes. Failing that then using software like Recuva is a good way to go. As a last throw of the dice you can try putting the hard drive in the freezer, (put it in a plastic bag to prevent condensation), it may get the drive working for long enough to recover the data.
If you have one available it would also be worth trying the hard drive in a different caddy.