And the solution? Putting the drive in the freezer!
I enclosed the naked drive in a plastic bag to keep out moisture and froze it overnight. Then in the morning I let it warm up for several hours and plugged it in. I let it run for about an hour to get it warm again (like drives normally get when running; remember the drive would spin but not mount). And it mounted, and ran fine! And I was able to copy all 30 GB of data to a new drive. The theory being that the freezing caused parts to contract and the rewarming caused them to re-expand and this release the stuck parts (probably the read/write arms).
[kirkmc adds: This hint follows another hint about cooling an iPod to get it to work that was published a few days ago. Several people mentioned similar techniques in the comments to the previous hint: either putting a drive in the freezer, refrigerator, or simply on a cool-pack. While this sounds like hard-drive voodoo, I'd certainly try anything if I was in that situation.]

