It’s possible the connector to the screen is just loose. As a last-ditch effort, you could try to repair it. See if ifixit has a guide to taking your kindle apart. It should be easier than a phone since there’s no waterproof sealing.
Just reseating it could fix your screen. Just make sure to be careful with whatever little black tab locks the ribbon cable in place. Some flip up while others slide out and they are notoriously delicate if you pull them the wrong way.
If that doesn’t work, plenty of eBay sellers offer replacement screens. I know this because my job had a bunch of left over e-ink displays from a product we were developing. They happened to be the ones used in the Kindle XL and I hocked them on eBay to a repair shop.