I don't really follow what you are trying to do - you are connecting wirelessly to someone else's wireless network (which is illegal) while your laptop has wired connection to your own wireless router.
<<< What I am trying to do is connect the desktop PC to the laptop via the wireless router so I can use its Internet connection >>>
That suggests you want to use your wireless router to connect the PC to the laptop to use its [the laptop's] illegal wireless connection.
What you should be doing is to connect both of your computers, either by ethernet cable or "wirelessly", to your own router. There should be no direct connection between the two computers which will in any case be networked via the router.
To connect wirelessly to your own router, its wireless capability needs to be switched on and broadcasting of its SSID enabled.
You then need to "Connect" to the wireless network? In other words, you run the wireless adapter's software, scan for "Available Wireless Networks", find and highlight the one that matches your router's SSID (network name) and selected the option to "Connect" to it. Once "Connected", the wireless adapter should get its IP address by DHCP from the router and you should have network and internet access.
There is a difference between the Wireless Network Adapter and Router communicating with a good signal and "Connecting" (as in "logging in") to your Wireless network.