It just turns on and powers down right away. It has a blue backlight. what is causing it to shut off?
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Alive and well in fucksville
tried newly recharged batteries
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Michigan
What is the mAh rating of your rechargeable batteries? Try four copper top duracells.
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Seattle, WA
If it isn't batteries, try temporary de-soldering the backlight's leads, wrap the tips with a bit of electrical tape so they don't short anything out, and try it again. While your at it, make sure you didn't accidentally short the board at the point you soldered the backlight.
I'm assuming your gameboy worked normally at some point in time, when did it stop working? Was it right after you first installed the backlight?
If it's a short, check for any part of the board that is unreasonably hot while the switch is on.
Try these things, and we can continue trouble shooting if nothing fixes it.