Sounds like a hardware failure (ie: controller board) in the hardware itself or the interface in the enclosure. If it were bad sectors or whatever on the disc, chkdisk would be making the repairs. I would get another HDD to copy all of the data onto before it completely fails. You may have to copy some contents and then shut the drive off for a period of time for it to cool, then try again. Or simply keep disconnecting, power off, then connecting and power on multiple times until you can get all of your data.
After you have your data safe, if it's under warranty, contact the manufacturer. If not, you could try removing the drive from the enclosure and directly connecting it to your PC. That will at least help in identifying if it's the drive or the enclosure.