Ignore the light.
The light (and the battery management) is partly under firmware control.
Your firmware is known to not be operating normally, so can you thrust the light?
The battery management (which includes the charger control) is partly under firmware control.
Your firmware is known to not be operating normally, so can you trust the charging controller?
Hook the device to a known good USB cable, using a known good wall charger, plugged into a wall outlet that does not get turned off by the light switch when you leave the room.
Leave there, hooked up, for a couple of weeks.
Yes, that word is:
weeks, it is not a typo.
Why?
You have to presume that the charging controller is defaulting to the minimum trickle charge;
and
The battery is old, probably more 'rocks' than charge holding structure, it will be more likely to heat than to recharge;
and
There are boot time tests (such as file system tests) that get run and internal repairs attempted at each re-boot. Each time a change is made, these have to be restarted from the word go.
The eMMC device is not the fastest thing in the world, it can take a few weeks for the self-fix code to finish running.
and
The eMMC used has an internal controller, rather simple but "smarter" than a flash disk stick, not as "smart" as an SSD. Like any eeeprom device, it does not support a 'write' operation. The internal controller is faking a write operation out of an "Erase-Merge-Reprogram" sequence. When running one of these internal cycles (and there can be a lot of them running a file system repair application) any failure (including shortage of capacity) of its power source can cause that micro-controller to "get lost", locking up the process, never to be good for anything again except as recycled beach sand.
and
The K3 has a recognized failure mode, for totally unknown reasons, that is exactly as you describe. This failure mode can not be recovered from, sorry about that, it is a built-in feature of the K3.
For all of the above, the correct word is indeed:
weeks.
And don't expect successful results.
This is just the best "first thing to try".
In the mean time, use the sticky "where do I start" master index to look up how to connect to the K3 serial port.
It will most likely be required.
While doing all of the above, keep in mind that Amazon is offering a trade-in allowance of from $5 to $10 on a K3. Yours is at the $5 end of the offers.
You could probably e-bay the display for about those amounts.
As used beach sand (silicon) that has to be re-ground - it isn't worth the time and effort to throw it away.