Macbook will say it's charging, but it's not
I finally fixed it today. It turns out the Magsafe DC-in board can be replaced. It's a little $25 part ($15 refurbished). I followed the instructions on iFixit.com; you have to take apart practically the whole laptop which includes a lot of tiny, tiny connectors, but it's doable with just a few screwdrivers and a spudger I made from a chopstick.



The battery now charges fine, and the laptop doesn't get nearly as hot (that could also be because I cleaned the heat sink).