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Old stock batteries are poorer than new made. The charge capacity of Li tech drops with use and age. The actual cycles don't count if you don't go below 30%, just total AH delivered and replaced.
Was the new battery reading over 3V? I'd not expect a new one to give a fully charged voltage as they are usually stored at about 2/3rds capacity, about 3.4V.
You can manually charge unplugged using a current source, or a resistor to limit current from 5V to about 100mA. You need to monitor temperature and terminal voltage. Stop if more than 40 C or 3.6V.
This will tell if the problem is the charging circuit has failed on the Kindle and if either battery is OK.
A dead battery will heat without charging. Eventually it would go on fire, which is why power circuits on anything with a lithium battery are usually designed to switch off/disconnect if battery is too low and not attempt to charge if the battery seems to be at the end of life.
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