Can't say l know what the problem is with your Kindle but if as you say it won't turn on you probably need to reset it. I found the following info on the Amazon site.
Press and hold the Power button for seven seconds until the Power dialog box appears, and then tap Restart.
Note: If the Power dialog box doesn't appear, press and hold the Power button for a full 40 seconds until your device restarts.
Beware: if you reset your Kindle you will lose all your downloads. If you have all your content on your Amazon account you can easily download it again should this happen. Let me know how you go.
There is lots more troubleshooting info on the Amazon site.
My interest, I also own a paperwhite Kindle.