You say your computer is connected wirelessly and you want to connect the Wii to the computer wirelessly?
If you can connect wirelessly to the internet with your desktop, you should be able to connect the Wii in the same way. It has built in wireless adapter.
You must have a wireless router that your desktop is connecting to ?
What you have is a wireless link. Generally that is plugged into your desktop which always needs to be connected to the internet wired. The link turns that into a wireless signal to which you then can connect the Wii to wirelessly. Typically these are for people that do not have a wireless router.
If your wireless signal reaches your desktop ok you could move the Wii closer.
The signal from the dongle will be too weak for the Wii unless the desktop is hard wired to the internet. . One bar wont cut it for the Wii.
In the Wii manual it tells you that you need DSL/Broadband/Cable. Even High Speed Satelite doesnt work for most people. You need 3 full bars in the Wii settings in order for the Wii to operate online properly. Also check to be sure you choose the same security in the Wii settings as the router has. (there are 4 types). Also check the SSID it must also match exactly.
WEP VS WEP Personal etc.
Nintendo discontinued the USB WIFI adapter and patched alot of generic lan adapters. I used a generic for awhile then they patched and I had to buy an official Nintendo Adapter. Im sure some still work.
It would help if you posted the error code you get. Thats why they give them so you have an idea whats wrong.
Another problem is anytime you change the settings you almost always have to turn off the Wii then reboot the router for it to work. I learned that the hard way.
Your other option is buy A USB LAN Adapter and go wired.