I just googled your monitors specs, and it seems it may have Component ports (I will try and confirm this), which means you wouldn't even need a VGA Box, just the Nintendo Wii Component cables. Also, the VGA Box you linked to has Component input, so you would need the Nintendo Wii Component cables to use it. The regular Composite cables would not work with that VGA Box.
Finally, I should warn you to be careful with your selection of VGA Box, as I have heard some (or many?) produce a bad picture quality. You should try and find reviews and stuff to ensure it will produce a nice picture quality.
But yea, I think you won't even need a VGA Box , just the Nintendo Wii Component cables, but I will try and confirm this for you. Well if you see 3 RCA connectors on the back of your monitor, then I'd say they are the Component inputs .
Sorry, I'm now quite sure your monitor doesn't have Component ports :\. So yea, you will need a VGA Box, unless you somehow modify a Nintendo Wii Component cable into VGA (hd-15) cable.
Ohh, and I think if the VGA Box you get doesn't require power from your PC, then it should work okay without your computer switched on.
Finally, I thought you may want to read a post I made in this thread, about the picture quality on LCD based displays.