Your external drive has two parts/devices, a USB/storage "controller", and an actual hard drive.
The controller in the external enclosure handles the interface between the drive in the enclosure and the USB connection you'll use to attach it to a computer.
If the controller still works yet the drive itself is dead, you'll see the USB controller as a USB device, but no drive will show up (because the actual drive in the enclosure is dead).
So IMO your drive is dead, either replace the whole external unit or open it up and replace the drive inside.