I've seen this type of situtation several times before on external USB/FireWire drives when the external controller glitches or fails. If after removing the drive from the enclosure and attaching directly via SATA/PATA, it spins up and is recognized by the system but does not appear as a drive letter than that normally indicates that when the external controller failed it also damaged either the MBR (Master Boot Record), VBR (Volume Boot Record), or both. The good thing is that most if not all of the files should still be there.
In this case since you say it is a backup then you would still have the original files and should not need to recover anything. If there are no files that need to be recovered the drive can simply be repartitioned, reformatted, and then either used as an internal drive or installed into a new enclosure.
If you do need files from the drive then you could try using something like Recuva
or other data recovery tools
to either repair the damaged boot sectors or simply copy off the necessary files and then reformat. Let me know how things turn out, Scott.