3.7volts isn't perfect at all, BTW. I rather guess the battery's dead and the PMIC or the SoC is rebooting the system due to brown-out detection. Can't add much more here, but I really suggest to go and try to recharge the battery for _hours_, up to 24, I'd say. With a crafty charger, not a PC connection - the Kindle will eat up all power otherwise.
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