Is the Wii softmodded? Do you know what caused the white screen? Did you install any wads prior to this happening? It sounds like you have a bad IOS. Since you can play games, try using the lego indiana jones pawns hack to fix it.
Post cvs report here if possible. If the Wii is softmodded with the system menu IOS patched, I can also send you a file to burn to DVD that will run Dop-Mii which you can use to restore.