In desperation, I turned to ebay. Luckily people sell them there. I ordered one ($4 + $2 shipping). While I was waiting for delivery of the screwdriver I kept trying to make a screwdriver out of a nail head using my Dremel. I still hadn't been able to do it after 5-6 tries. These failures turned out to be helpful practice however because the screwdriver I ordered from ebay was so cheap, the tip broke off the first time I used it. So it was back to the Dremel. 2-3 more tries later and I had finally gotten good enough to fashion something that worked.
And you know what I did after I got the screws out? I slapped a cutting wheel on the trusty Dremel and cut a straight groove on the head of each screw. So if I ever have to open that fucking thing again, I can just use a standard screwdriver.
Eat me, Nintendo.