Unless a drive formats correctly it wont have a drive letter alocation (G:, H: or whatever). So it will not show up in "Computer".
Because an external hard drive has both the actual hard drive in it AND some extra electronics it is always difficult to know if it is the hard drive that has gone wrong, or the electronics.
The hard drive MAY be fine, but you will need to take it out the case and put it in another case and see if it works.
Because you were using this external hard drive for backups, then as it is so unreliable I would not risk it for backups in the future anyway, even if you can get it working.
May be time to buy a new external hard drive.