If you are sure then,
- Windows sees the device connection because the electronics board inside the external drive case communicates with the operating system but either that board is partially malfunctioning so it fails to establish a connection to the hard drive hence Disk Management not showing it.
[ OR ]
- The board is fully functional but the hard drive inside the case is not functioning properly or at all.
Your best best is to open the external drive case and see if there is any humming, vibration coming from the drive. Also check for any kind of clicking/grinding sounds which would indicate a serious internal problem.
If the hard drive has absolutely no sound or vibration then the only way to be certain the drive is bad, would be to connect it directly to a working PC's main/motherboard and check.
[ OR ]
As mentioned before, you can take the drive out of the casing and take it to a local computer shop/tech savvy friend, who could check it for you, perhaps even for free.
PS. Check the USB cable connection at both ends (external drive end and PC end), to make sure it is firmly connected. If possible, connect the drive to another USB port on this machine to rule out couple of things.
Even better would be to connect it to another -known- working PC to double check.
Always start with the simplest, easiest and cheapest troubleshooting step and go from there AKA low tech before high tech.