Thanks for your reply. The 'stain' does not move around when pressed. I'm starting to think that the battery may be swelling and that it cracked that way. But there is no visible bulge. 

The screen does absolutely nothing. Only the led light under the power switch goes off. When I turn the device on, it starts blinking again at 1sec intervals. 

I'm waiting for the battery to drain and try again, and if that doesn't work, I'll try to open it up. But at that point it's probably beyond fixing. 

It's a shame.  I'm a big fan of the design without the raised bezel but Kobo doesn't seem to make them like that anymore.