SATA or PATA????
Have you tried to remove it from the ext case and installed it into the computer...to check if any of the electronics in the external case failed.
could be something with the usb drivers
in the new machine... might try unstalling from the device manager the usb controllers, maybe the composite device then turn the machine off and let windows reinstall 'em again and see if the drive is picked up.
At this point I'd be asking just how important is the data on this HD... I'd even go so far as see if I could put the HD into a friends machine,(or old 'puter), to see if it could be picked up???? Be sure and copy the data off the HD while you continue to trouble shoot the drive.
From experience..... the only drive I've not been able to finally detect ended up as a dead drive, "bfh"
luk to you.....