Hello World,
It was hard to find a good picture of the battery contacts inside the 3DS, but it looks to me like they're just spring contacts and I'm guessing they've gotten bent inward too far such that they aren't making contact when the battery is in it's normal position. If it was mine I'd try taking a thin metal pick and insert it into the loop formed by the contact and bend it outward. Once bent, it should remain far enough out to touch the battery after it's been placed into the housing. You may have to play with it a bit to figure out how far it needs to be bent, but that looks to me like all it'll take to get you going again.
Good luck, and let us know how it goes.