It was clear that there was a broken or barely hanging on electrical connection in the power jack.
I put this off a while, thinking it would be hard, but it turned out to be a really easy repair.
You can buy replacement jacks on ebay and amazon for <$7 including shipping. Make sure to buy it somewhere from a US vendor, not from a vendor shipping from China/Honk Kong. You would still get it, but it will take you weeks.
Found them on amazon.
Power Jack Socket Dock Connector Charger Charging Port Nintendo 3DS Replacement
Bought this replacement jack off ebay for $6.35, free shipping. Came in two days!
Watched this video. Not bad this kid shows you how to get the cover off.
how to fix nintendo 3ds charging port - YouTube
You need a really small Phillips head and optionally a plastic pry tool. If you are careful or use your fingernail you don't need the pry tool. You can buy them on eBay for pennies.
After the 4 screws on the battery cover, There are nine screws and one on the game slot you have to take out. They are very small and were in really tight. A good screwdriver is needed, and press hard.
Take out the SD card and battery and any games and the bottom half of the DS pulls right off.
Watch out for the flex cables, just leave them connected and don't pull them.
I found the power jack actually just sitting in place. All of the solder connections were broken loose and I could just pull it off. The new connector went right on top of where the old one was.
I used a soldering iron and solder to attach the 6 tabs to the pads. It's obvious how it goes. My solder job was crap because it's small and my iron was big, however it was still no problem.
Put the lid back on, screws back in, battery, SD card, cover. Powered up. Ba da bing. Plugged in the charger. Yellow light. Win!
This is a really easy repair. If the connector had not been obviously broken, I would have had to use the soldering iron and the little vacuum desoldering tool to remove it. Available at radio shack. Luckily I didn't have to do any desoldering.