In disk management, right-click on the drive and choose Properties. The on the hardware tab click the Properties button.
Now you have a window with information and tabs. Click the Volumes tab to see what's there, clicking the Populate button if nowt comes up.
See what that does. If there's an actual problem with the drive this should indicate as such. Failing to unmount the drive before unplugging it shouldn't frag a drive though, just potentially lose data.
have you tried the external HD on a different PC or laptop?
If it does not come up on another computer - it is very possible that the disk controller is fried.
If you need the data on the drive
carefully open up the casing (some casings are easier to open then others) and remove the hard drive (taking Anti Static Precautions) - buy yourself a disk caddy from a good hardware store and place the disk in the caddy - you should now be able to get to the data on your hard drive.
There are 2 type of drive used - SATA or IDE - the IDE drive has several small pins sticking out the back.
You can buy disk caddies that can handle both, getting a single type caddy is cheaper though. Newer hardware should have SATA connections