After a few weeks of tinkering, the wireless is finally working in Ubuntu on the laptop my aunt sent me.
My new (old) laptop is a Dell Inspiron 9200 with a Broadcom 4306 (rev 3) wireless card. In theory, all I needed to do to get the wireless working was grab the firmware and activate the drivers. Despite nothing appearing to go wrong during that process and everything ‘looking’ right, I couldn’t even scan for networks, let alone connect to one. So, after reading approximately 45,917 forum posts, 17,337 how-to guides, and tinkering a bit, I gave up on using the Linux drivers.
Then I found this how-to guide on Broadcom 4306 and Ndiswrapper and Windows drivers for my card here (via davidwatson.org). Approximately 6 minutes later my wireless is working and my laptop is no longer tethered to the router.