This morning I have found that even after recharging the battery voltage was 3,1V. So both of my batteries are dead and either the new one was also bad or Kindle is draining it due to some hardware problem.
Can I take the ordinary AA batteries (3 pieces = 4,5V) and solder the wires to + and - in Kindle, just to test it? What to do with additional 2 contacts in Kindle?