Sorry I am not sure I fully understand what Linksys device you have and are replacing.
If you have a new USB wi-fi dongle then you shouldn't need any other hardware, assuming you have a free USB 2 port. You should just be able to use the usb dongle to connect to the Range Extender in the lounge.
Assuming XP, in Control Panel go to Network Connections icon Click once on the Wireless network connections icon and then from panel on left select View Available wireless networks. Depending on the network security settings (WEP) you should then be able to connect. If WEP has been set up then you will need to set the desktop up using the same key.