There isn't a switch or the like to reset your Kindle from the motherboard. You could buy a serial adapter and temporarily solder (or even tape if you are careful) it to the motherboard. This would give you a lot of information about what the Kindle processor is doing and a way to reset your Kindle by typing commands on a connected PC.
You would need a 1.8v TTL serial to USB UART adapter e.g. like
this. I have also seen them priced < $3 from Chinese vendors on ebay. Although you get more erratic quality control and documentation and have to wait a month or two to receive items from such vendors.
What type (chemistry) and voltage was this cell phone battery that you used for the Kindle? That could easily have a lot of bearing on this situation. Something that you might have mentioned much earlier.