Understand that an external drive is really two components, rather than one. So, if an external drive doesn't appear to be working, it could be one of two things: the drive or the enclosure.
First thing I'd suggest is to remove the drive (the real hard drive) from the enclosure it's currently in. The warranty period has expired (1 year) and there's nothing to lose.
Then, I would connect that to the system as a secondary hard drive, ensuring that the jumpers match up with the jumper setting of the other drive on the same cable connecting to the motherboard. If there is no other drive on the same cable, ignore the jumpers and just connect it to the end connector.
I would then check that the drive is properly recognized in the BIOS.
I would then reboot the system and see if the drive is properly detected in My Computer. If it is, great. If not, then I would check Disk Management and see if it is reflected there.
If it is not reflected in Disk Management, I would start thinking "data recovery software."