It was a rough week for my Surface RT, apparently, as it decided to stop recognizing its own battery in an existential sort of way. Very meta. Good timing as well--one week past warranty.
(When on, it showed a battery error icon in the tray. Once the battery died, the charger light didn't turn on when plugged in)
Firstly: Ebay is full of spare parts from sketchy vendors. Luckily, I decided to unplug and re-plug the battery and charger port first, before hastily buying anything else. Worked great!
I used this guide: Microsoft Surface Charging Port Replacement
Took about 45 minutes total. The hardest part was removing the plastic camera cover with the spudger. The tolerances were too tight to fit the spudger into the crack and I ended up *very carefully* using a knife to work the far right edge up (looking at it from the back). Once I got that up, it was straightforward to carefully pry the camera cover up. It is fairly flexible and even though it warped a bit during removal, it never broke and was easy to bend back into shape during re-install. There was one tricky spot, though: unlike the "tabs" around the edges of the cover, there is a "post" that holds the camera cover in place immediately to the left of the hold button. It must be lifted directly up (not "back") to be removed. This took some force and placing too much leverage on it from the rest of the cover resulted in slightly creasing the cover.
I think I covered most of it above.
Use a couple magnets to hold onto the screws for you while you work. I used a magnetic paperclip holder. There are several different lengths and some of them have locktite on them, so try not to get them all mixed up
Don't be afraid to try this! My friends all thought I was crazy, but I was careful and confident and it all worked out.