Amazing! I got it, guys. While I was typing the info from the back of the card, the name "Agere" rang a bell with stuff I've read on this forum, so I did a search here for Agere, and I found one chap who mentioned that he had used the Agere driver with a similar-sounding card. So that's what I did, and it worked like a friggin charm!
I got the driver here:
http://www.agere.com/mobility/wireless_lan_drivers.html
just click on "Driver for Windows CE only" -- this gives you a zip file with three folders, one of them being "CE 2.11". So I used that one, ran setup, and voila! It installed it to the machine, and it was even nice enough to "hold my hand" and tell me to remove the device from its cradle before inserting the card... okay!
So I inserted the card, and immediately the driver config panel came up, and it started searching for access points. I figured it couldn't find my home network since I have WEP enabled, so I quickly went to the PC that houses the router, opened the wireless security panel, got the key, and as soon as I keyed in the key and clicked OK, something happened, but the system froze... but a simple soft reset later, the card was already connected, with an "Excellent" connection! And web browsing was instantaneous...
Thanks for listening, guys, and I hope this will help some other people around here as well.
Harris