Mount the drive read-only and see if you can get the files off it:
sudo mount -t ntfs --read-only /dev/sdx1 /mnt
I'm assuming since it's a Windows machine that the partition is NTFS and you'll need to change /dev/sdx1 to the correct partion. You can verify both of these with the command.
If you've got an external backup hard drive with enough space you could do a bit copy of the entire hard drive to a file on the backup drive eg:
sudo dd if=/dev/sdx1 of=/path/to/filename
This will take a while of course. You can keep tabs on the copy if you open up another terminal window and type
watch ls -l /path/to/filename/
This is SOP for data recovery as long as you have the extra disk space (normally you would want to use a SATA drive because of the copying time involved). You could then mount the image to copy the files off it with no fear of the drive failing further, or you could run testdisk on it to recovery missing files.