I think this is a rare case but maybe it'll help someone.
I have a kindle keyboard whose battery life was finally getting very short. A few months ago it stopped working altogether: the screen went weirdly dark gray and the power switch did nothing, no matter how long you held it. Plugging it in got an orange charging light that would eventually turn off. It never turned green.
I thought about replacing the battery but the screen, besides being a shade of gray I'd never seen, had one bright white horizontal line and I thought there might be some damage there as well. In which case repair might not be worth the cost.
Everything I saw online just said to charge longer, use slow chargers, play around with the power switch, etc. Finally I found a blog post about fixing a dead kindle by short circuiting it. I thought I didn't have much to lose so I opened mine up to try - and then as soon as I plugged it in, without short circuiting or anything else, the light went green and it turned on (and the screen was perfect with no dead pixels).
There was a tiny screw that fell out when I opened the case. It must have gotten knocked out of place at some point (maybe the kindle was dropped recently). So there's a chance that screw had wound up getting in the way of something. I also found some
references online to a micro-switch getting stuck and to a screen cable getting loose. It's not out of the question that the jostling needed to open the case got something back to its proper position.
The moral of the story: if you aren't able to get your kindle to turn on and are at the point where you think you'll have to get rid of it, at least try opening it up and taking a look.
I'm not sure if the problem's solved or if it's just a temporary fix - the battery is nearing the end regardless - but at least now I know the screen is fine and getting a new battery wouldn't be a waste.
(And I was inspired by this new lease on its life to finally install koreader

To NiLuJe and everybody else who's made it possible to do that - a million thanks!)