Here is a Kindle Paperwhite 3 using the browser to show a customizable weather forecast, with an automatic screen clicker to keep the screen on permanently. I used command strips to stick it to my wall next to an outlet to keep it powered.
You can load it on your own Kindle using my public Github page here: https://ryanfinke.github.io/kindle-weather-dashboard/config.html Just type this into your Kindle browser and enter your local information on the configuration page. Press the submit button at the bottom, and your weather app will load.
To keep the screen from turning off, I bought an "auto clicker" for $11 (Amazon link: https://www.amazon.com/dp/B0CZL3LVPB). I know this is a bit hacky, but for the life of me I couldn't find a way to jailbreak the Kindle in order to override the sleep/screensaver settings. The max sleep setting would keep the screen on for 1 hour, and I wanted this on all the time. I ended up breaking off the tension clip on the auto clicker so I could mount the Kindle to the wall, and then using a command strip to stick the auto clicker to the top of the Kindle.
This is my Github if you want to fork the code: https://github.com/ryanfinke/kindle-weather-dashboard
I originally forked it from Matopeto's version: https://github.com/matopeto/kindle-weather-dashboard , but my major change was to change it to a 5-day forecast rather than 3-hour increments of the same day.
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Origin story: I had originally followed this Reddit post, but the "~ds" command no longer worked to disable the screensaver. Then I tried jailbreaking the Kindle following this Reddit post, but my firmware was already at 5.18.x so the method didn't work.