Chosen Solution
I got a new battery from ifixit which I thought would do the trick, but the symptoms are still the same: when plugged in it gives the “battery” icon (unchanged for days at a time), and the screen basically stays on with a flicker to a dark battery every 20 seconds.
I have a similar issue. You might want to try plugging it into a USB port on a laptop or PC/Mac. The low voltage charge from USB2 ports can slow/trickle charge and wake up the battery.