After only a few minutes of tinkering, I'd fully rescued all of the data from my old drive. Here's what I did, and you can, too!
First, get out some tools.
You really only need a pocket knife, but the Leatherman Crunch looks so cool. And I put a screwdriver in there because using a screwdriver instead of a pocket knife will lead to not stripping as many screws.
Use the pocketknife to gently part the lid from the plastic. It's held in by glue, and all you need to do is wiggle it a bit on each edge to get it apart.
Or, if you're like me, pry that sucker open like it was filled with Raspberry Jelly!
Alright, see that cable that you just accidentally unplugged? (Unplug it if you didn't)
Remember that. you'll plug it back in later. That takes power from the USB connection and lights up that emblem on the front of the case.
Now use your pocket knife to pry the rear side out.
It's not hard. It's held in by friction on those little rubber nubs on the side of the drive. If you flip it upside down and shake it, it will fall out. Note that shaking anything computer related is not really advisable. Just pry it out, slacker.
Remove the black nubs from the screws. They just fall off, usually. They'll roll under stuff and hide if you're not careful. You'll want them later, when you put all this back in the shell. So put them somewhere so you'll be able to put them back on, in a few minutes.
Then use a screwdriver to remove the screws holding the drive in the metal shell.
Now remove the metal tape from the sides of the drive, then remove the drive from the shell.
It just slides up and out, with the connector port poking through the shell.
And as a final step, remove the connector from the front of the built-in drive.
Now plug that connector that onto the old drive from your laptop, and follow these steps in reverse, to put your old drive in the portable drive's case.
When you're done, you'll have your old files safely stored in a portable external drive case!