Several mobile devices switch to battery power for a moment while powering on. If the battery is dead then when they do that they just power right back off. I've seen it happen on older Android devices, which share the same/similar chipsets as used in Kindles. If you were comfortable working with the electronics you could remove the battery and clip wires from a bench powersupply set to the right voltage to the terminals on the board. That would let it run as if it was on battery power and at least confirm general good working condition.
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