Have you check if the hard disk works on different ports on your computer? If the PC has several USB ports, I recommend plugging it in each of them because, for example, the USB ports of the front panel of the PC may need updating of the drivers. Besides, I would like to know, have you connected the external hard drive on different computers with different operating systems recently? I am also interested if the external HDD has additional power or receives the power by USB cable. I'd appreciate if you give me more details about that.
If you have the option, I recommend you go to the Control Panel> Computer Management> Disk Manager to tell me if the external drive is recognized and has a drive letter assigned to it. If it hasn't you must assign it manually. It would be good, if seen, to test it with a software made by it's manufacturer to check its S.M.A.R.T. Status. If you haven't done it till now, I would like to suggest you try to access it with an Ubuntu Live CD/USB.
If the hard drive is not seen, you can reboot your machine while the disk is plugged in and enter the BIOS menu to see if the system can recognize it 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 you can't, the disk must be in a bad condition and you should take it to a data recovery company to extract the information.