Sounds like it isn't getting a proper connection somewhere. From what you've described and tried, it looks like it can't be the connector of the charger or the port on the system which is pretty odd.
Try reseating the battery and see if that works. Remove the back cover, take and leave out the battery for a short while, then put it back in. If that doesn't work then the only other thing I can think of is getting a new battery from Nintendo themselves on their website.
OT: The charger for the DS and the 3DS is the same? I thought they were different. Anyways, try using a different outlet.
The DSi and 3DS use the same charging port while the DS Lite and DS use a different port. I'm going to take the guess that Nix was referring to his DSi when he said regular DS.