From GM's description, it could be almost anything on the storage side of the board recovered from a dead Kindle battery.
Say for instance, two, 2.7 volt rated capacitors in series (rated maximum voltage 5.4 volts) that never get charged beyond 4.2 volts by the Kindle's battery management.
Any pair of the ones described in the attachment would be interesting to try.
1,000,000 charge-discharge cycles (many more when not charged to the limit) -
minimum 10 year life (probably more in this sort of application - since the Kindle battery management will not completely charge or discharge the capacitor pair).
Take off from Kindle - put on external charger - recharge them in under 20 seconds.
A capacitor diameter of 60mm is going to make a noticeable lump in the zipper case.
An interesting experiment - someone give it a try and write it up.