Tool to copy files of HDD with bad sectors
Roadkil's Unstoppable Copier recovers files from disks with physical damage and allows you to copy files from disks with problems such as bad sectors, scratches or that just give errors when reading data. The program will attempt to recover every readable piece of a file and put the pieces together. Using this method, most types of files can be made useable even if some parts of the file were not recoverable in the end.
Unstoppable Copier is freeware.
If that doesn't help, try fixing the drive with HDD Regenerator and salvage your files.
HDD Regenerator is shareware (US$39.95, money-back guarantee) and is try before you buy.
PS: I have pretty good experiences with this program, but of course success depends on the severity of the damage.
If you can handle Linux, ddrescue will do a block by block copy, recovering bad sectors, (like the dd command) to another hard drive. I have used it successfully in the past to recover the contents of a hard drive Windows wouldn't mount.
It is included on the Linux System Rescue CD so you don't need to install Linux on a hard drive.
I wrote a simple PowerShell script here which allows better fine-tuning than Roadkil's Unstoppable Copier. Precision is up to the cluster size.
Currently I am trying to access the RAW disk/file with it, so that I am less limited and have more recovery options.