Have you tried to plug the hard drive to different ports on your computer? If not and the PC has several USB ports, I recommend you plugging it in each of them because, for instance, the rear USB ports or that ones on the front panel may need updating their drivers.
Besides that, I would like to know, have you connected the external hard drive on different computers with different operating systems recently? If so, tell me more about it. I am also interested if the external drive has additional power cord or it receives the power by an USB cable. I'd appreciate if you give me more details about that.
If the hard drive is not seen in My Computer, you can reboot your machine while the disk is plugged in and enter the BIOS menu to see if the system can recognize at a physical level. Can you hear the hard drive spinning while connected or everything is calm and does not feel any vibration when you touch the case? If so, the disk must be brought to a data recovery company in order to help you extracting the information.
Please, share what happened.