Kindle number 1 and 2 were mine, number 3 belongs to my friends.

I bought two new batteries, to fix two Kindles.

Kindle number 3 was the most recently used, and had my friends' books on it. They wanted it to read on their travels, so I looked at it first. It managed to charge and wake up, but the old battery number 3 had died the next day. So the Kindle is working but the battery is dying of old age. I replaced it with new battery number 1, and it seems to be working now.

Kindle number 1 is my original. Old battery number 1 was at zero volts and swollen, so it is seriously dead.
I fitted new battery number 2, plugged it in to charge, but failed to get past the empty battery screen. Maybe something wrong with that Kindle? I moved the battery to Kindle number 2.

Kindle number 2 appeared to be happy with new battery number 2 for a while, but after a day it had a '?' over the battery icon. So I tried a reset by holding the power button down for 40 seconds. This made things worse, and it failed to get past the empty battery screen. Repeated resets did not fix it. The voltage is a very healthy 4.1 volts, so I am guessing it is having trouble with the battery management chip inside the battery. Maybe the OS needed updating, but I can't do that without booting up. So I fitted old battery 3 for that purpose. It managed to boot up.

It says the serial number begins with B00A and the version is Kindle 3.4.2 (2687240004)

https://www.amazon.com/gp/help/customer/display.html?nodeId=GKMQC26VQQMM8XSW offers several choices of files to download depending on the serial number. I downloaded the one for mine, told the Kindle to update, and it now says the version is Kindle 3.4.3 (3543620001). Sometimes fails to refresh whole screen when displaying images. 

Swapped the battery to new battery 2. Still thinks that battery is empty.

Fitted old battery 3 to Kindle number 1. Briefly lights up green, then blank screen. 20-second reset gets to orange LED, but stays blank. Same with 40-seconds. I left it to charge for a while but the orange light has gone out.

Swapped batteries back. Kindle 2 wakes with '?' on the battery icon. I did a restart from the setting menu, and it changes to a normal battery icon.

Conclusions:

a. New battery 2 is reporting it is empty via the I2C bus, despite having a healthy voltage.
b. Kindle 1 has problems with any battery.