Have you tried to install backlight without the switch or solder wires other way around?
Also, can you take some pictures of your soldering?
I desoldered the negative lead from the switch and took it out of the circuit completely. After re-connecting the negative from the backlight, I powered on the GameBoy but got more of the same thing--a whole lot of nothing. I have another backlight, so I quickly swapped it out to see if that would power on either. No dice.
Is there another way I can test the backlight to see if it's functioning properly? Forgive me if that's a terribly noobish thing to ask.
In my experience with troubleshooting, I can logically conclude that I've wired things wrong or that there is something not functioning properly in the actual GameBoy (or at least that something isn't occurring at the contact points).
EDIT: Nevermind. I have determined the backlight is, in fact, dead/dysfunctional. I tried installing it on another PCB with no luck so I tried the other backlight that I have on both units and it worked fine.