Regarding Nexus 10 and the Android 5 "upgrade", there's a long thread at
https://productforums.google.com/forum/#!topic/nexus/kMJx6MXXYTw;context-place=topicsearchin/nexus/%22nexus$2010%22,
where the conclusion seems to be that Android-5 somehow breaks charging, and that downgrading to 4.4.4 or earlier helps.
On my device, I had barely enough power to boot to fastboot mode and install a 4.4.4 factory image. But..., new software can't have any effect until it's booted, and it still refuses to boot.
I'm now considering following the instructions here to open it up, try resetting the battery, and if that fails, try to power it in some way bypassing the battery logic to get it to boot 4.4.4, which I hope would reset charging and battery logic to a working state.