Having worked on hand-held radio systems, I saw their battery packs included chips that managed the power cells. They would count the charge passing in and out, implementing the equivalent of a car's "gas gauge". 

I believe this has got confused, perhaps being cleared to zero and thus telling the Kindle there the battery is flat and not to start up.

One might be able to monitor the messages and decode what they are saying. This will need knowing what chips are used, and finding a data sheet.

2025-02-24

Under a microscope I read the chip markings. The one with the most pins (10) was connected to the I2C bus and was marked 27210 OBW Z332. Googling "27210 OBW Z332 battery manager chip" led to the Texas Instrument's BQ27210 here: https://www.ti.com/product/BQ27210

The data sheet https://www.ti.com/lit/ds/symlink/bq27210.pdf has a circuit for the BQ27010 which only differs in pins 4 and 5 which are named D/C and HDQ instead of SCL and SDA respectively.

The 7-bit I2C address is fixed as 1010101 binary, which is AA hex to write and AB hex to read.

There is EEPROM and RAM. The contents are fully documented in the data sheet.

A first step would be to read and analyse the contents. This can be done easily with a USB-to-I2C module:
https://hackaday.io/project/175889-i2c-interfacing-umft4222ev-module

A second step would be to monitor the I2C data traffic between the Kindle and the battery manager. This is harder to do, and requires a more complex and more expensive device such as the Beagle I2C monitor which is pretty expensive at $450. My workplace lab has one of these.