I know this is like 3,5 years old, but you may have purchased an old Kiosk/Demo 3DS (from a Techstore/Gamestop whatever). Those models can only turn on if theyre connected to a power outlet, which is planned behaivior by Nintendo, so that Thieves have less of an incentive stealing the devices.
How to fix it? That's the neat part - you don't. At least without soldering Skills, since those models have a Fuse "missing", which would be needed to use the actual charge of the battery for the device.