Oh yeah, good idea! Does that use the brass pads on either side of the DC jack? Could it be a fuse? I don't see how this one missing pad could cause this problem. It's just the casing of the jack - one of four ground pins.
Ok, what's the middle connector on the battery? I ran a wire to link the +'s between the DC in jack and the battery terminal. The ground pins already have continuity. I don't understand why it doesn't work. I've tried another battery. Maybe with this fix, the battery will charge but the LED won't light?
try using a cradle, it will solve one (or more) questions you are asking.
if the 3DS charges via a cradle, the fuses and battery are good, the DC jack probably needs to be re-soldered.
if the 3DS does not charge via cradle, does it charge with a different battery?, if yes, battery is bad.
if battery is bad, does a different battery allow it to charge via the DC jack?, if yes, problem solved.
If the 3DS does not charge on cradle with the original or a different battery, the fuse(s?) will need to be replaced.
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as long as the 3DS can at least charge via the cradle, you could always use it like how we used to play handhelds back in the Gameboy days
as in,
charge (equivalent to changing batteries)->play until almost dead->go to step one.
I guess the main difference is that unlike the Gameboy, you do not need to save and quit the game then turn it off, as you do not need to change batteries.
that is what I would do If i could not fix it charging outside of using the cradle.