Most of the screens we use don't support automatic source switching, and the few that do aren't well known to me. If your screen didn't come with a plug in remote control board, you'll have to solder a wire to a ground location and poke the screen control button pins with the ground wire to simulate pressing a button. The pins marked Source and + - should be all you need for that, but you could also try the Auto button.
As for the video signals, the RGB data signals from the Wii should be run to R+/B+/G+, and H-sync and V-sync should be run to HS and VS respectively. Those are the positive signal inputs. R-/G-/B- are ground locations for if you're using shielded or twisted wire. They're just there for convenience and can be ignored if you prefer.
Two things. The first is that you haven't wired H-sync and V-sync. Those data lines are essential for VGA , and you won't get any picture without them. The second thing is that many people wire VGA incorrectly because they follow the colours on the diagram rather than the pinout in the top right legend. The colours are for YPbPr component video, not VGA. Pin 11 is marked in red because it carries red data for YPbPr, but for VGA it carries B data. You may need to rewire that