If I were you, I'd get a LiveCD Linux distribution such as Knoppix, Puppy Linux, **** Small Linux, or Ubuntu. I recommend Ubuntu because it's most like Windows, but it's also a full fledged distribution, so it takes the longest to load. You can boot from the above CD's without any installation.
If you want, remove your desktop's hard drive and just connect the laptop one. Boot from the Ubuntu CD. Once you're at the Ubuntu Linux desktop, click the Places tab, then select Computer. You should see an icon of the laptop's hard drive. Click that to open the contents. Now you can go to Firefox, download the drivers to the Ubuntu desktop (in RAM), then copy the drivers to your laptop hard drive. Shut down the PC, remove the hard drive, put back in the laptop, boot up. Look for the NAME, not the letter, as Linux does not use drive letters like Windows does (it uses /dev/hda, b, c, d, etc for drives, hda1, 2, 3 for partitions).
I left this part out. I meant to tell you to do the exact same thing only I was going to say use a bart pe cd or erd commander 2005 to download the USB driver (you can do the networks driver too) to your hdd.