Have you tried it in a different machine? (Maybe one with USB 1.1 only.. the host interface is different (no EHCI for USB < 2.0) so it might work differently there. Of course the data transfer will be slow but that's probably not a huge concern if you are trying to retrieve the data).
Try unplugging everything else when you are doing it. Maybe there isn't enough power to support all the devices + the flash drive.
If that doesn't work, there's probably some kind of hardware damage, which could be due to various reasons, here are a few I can think of:
- broken USB plug (old drive = plugging it in and out a lot = stress on the connector)
- cold joint on the circuitboard inside (same reason, stress causes the solder points to stress and possibly crack, if components are coming loose from the board inside it's likely they aren't working. You could open the drive and touch some of these points to remelt and reconnect them, but I wouldn't recommend it unless you have the soldering skills / eyesight / patience to do soldering on tiny devices like these)
- burnt out / faulty components to start (not much you can do about this, except maybe "transplant" the flash chip into another similar / identical flash drive and see if you can get it to work... this is a gamble, but if the data is really important, it might be worth a try)
- etc.