So, even though it's over a year old, Amazon has been known to give a discount on a replacement device if you are willing to return the damaged one. I don't know if they recycle or repair or what, but it's worth a call to Kindle customer service to see what they can do for you.
What you describe as happening basically means that the screen has been damaged. It's not reparable -- unless you want to replace the screen yourself which, if you like to play with electronics, you can probably find out how to do with some Googling. No idea if it can be recycled as other electronics are -- you'd have to ask the place that accepts the items.
That said, the basic kindle is currently only about $80 . . .. $50 if you're a Prime member for a limited time. So it wouldn't be terribly expensive to buy a new one that would have a full year warranty.
Or, you might be able to get one cheaper -- or a higher end model for about the same money -- if you check "Warehouse Deals" on Amazon. They have a number of PapeWhite models on sale as well as the older basic kindles. I'm not seeing that they are directly sold by Amazon, though, so there's no warranty on them.