Try running it connected to a USB charger. Since it initially boots, that should provide enough power to eliminate the possibility of a weak battery.
It sounds like a good candidate for a motherboard transplant from a broken screen Kindle off Ebay.
[here]'s a detailed writeup by someone who did that successfully.
Don't take the whole K3 for disposal. With a good screen, its worth about $30 upwards for parts. Li-ion batteries that have been discharged past 2V/cell are not to be trusted, and disposal may be the correct choice for the battery once removed from the unit, but as you don't know if it ever got that low and it seems to take a charge, I'd keep it for now and see how long it runs a working K3.