Even if it does not look like it is charging and even though it may be fully charged charge it using a different wall outlet than you normally do (a different room is best) for at least 10 minutes.
Do not skip that even if it is fully charged from you already plugging it in, it is important to do it.
It sounds like the firmware may be frozen try this and let me know the results hold the power switch for at least 60 seconds.
Try this at least 6 times.
The firmware can sometimes get frozen the same way that a computer will occasionally.
Thanks Freddy it worked. The Kindle started up again and I will try charging it in another room although it IS fully charged.