After exchanging e-mails with Dell support confirming that I had done all of the trouble-shooting that their computer scripts say I should do, they've agreed to send me a new wireless card. I hope that fixes the problem. There's a possibility that it's not the wireless card but the antennas, which as I understand it run up through the top of the case. If they're broken, it's probably a more difficult fix than just popping the wireless card.
It occurs to me that I could test this by swapping not the hard drives but the wireless card itself.