Thanks for the tips!
Strangely, the problem seems to have solved itself. I left the device plugged in all day, and came back and the green light was on indicating fully charged. It then happily turned on.
I can't explain this behaviour, particularly why the amber light didn't come on while charging. But I suspect it was out of battery. Which is worrying as it had been fully charged just a day before.
I'm hoping that this is not a late occurrence of the kindle-case-discharges-your-battery problem.
Thanks again.
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