Is it possible to connect via Ethernet cable to the Router? I'm thinking that your LAN network adapter is probably working. Then SlimDrivers would work for you. You might need to move the desktop and monitor to a location near the Router.
Most Desktops do not come with a wifi adapter as standard equipment. Did you add a wifi adapter at sometime? It probably came with a software disk that included the wifi driver. Do you know the make and model of the adapter. If so you could download the software on another computer and copy it to a flash drive and install the driver via that drive.