Oh god, the Nintendo Wii USB Wifi Connector. I had one before I got wireless internet in my house, and it was a nightmare to get up and running.
First thing you HAVE to make sure you do is run it as an administrator (right click and hit "run as administrator"). If I remember correctly (as this was several years ago), it solved the vanishing taskbar thing for me. If it doesn't, try connecting the wii to it anyways, it's worth a shot. Make sure to put it under connection 3 on the Wii, it seems to work better for some reason. If that doesn't work, make sure you've got the right drivers for the right system (32 vs. 64 bit). Another possibility is to get a wireless router, it's so much easier
But seriously, another thing you can try is right clicking the taskbar, hitting properties, go to the "Notification Area" tab, and uncheck "hide inactive icons". If that doesn't work, try uninstalling then re-installing all components of the software. If THAT doesn't work, then I'm out of ideas.