I think it’s a common fault on the Paperwhite. I’ve just taken mine apart and replaced the power button flexistrip; not really a job for anyone who doesn’t dabble in electronics regularly.
But before that, I used this ‘workaround’: Plug it into a computer via USB. On mine, this bypassed the power button and it came on. I then ‘disabled’ the sleep mode so that it stayed on by entering ~ds in the search box. By turning the brightness down when I had finished reading, it stayed on for quite a few days before needing re-charging but at least I could read my book! If it went flat, I could just repeat the process.
The ‘button’ appears to be a very thin piece of ‘foil’ behind a gel pad which is pushed in by the black rubber button that is visible. The foil, when pressed in, touches two metal pads thereby activating the circuit. In mine, there appeared to be some black residue or dirt between the foil and the pads so they were not being connected. Some Isopropol Alcohol spray cleaned it sufficiently to make it work but then I managed to destroy the flexi-strip so had to fit a new one.