To get a Leanpub book onto your Kindle, please follow the instructions below, which are also on your Kindle Settings page.
Note that this only works if the Leanpub book has a MOBI version. Most Leanpub books have a MOBI version.
[On a side note: here are some instructions for adding PDF files to Kindle: https://www.pcworld.com/article/2044412/convert-a-pdf-for-kindle-viewing-no-software-required.html]
First, Get Amazon to Allow Leanpub to Put Books onto Your Kindle
First, go to Amazon and let them know that it's okay for us to send things to your Kindle. Personal Documents Settings on Amazon and add firstname.lastname@example.org as an approved email address. (Look for "Add a new approved e-mail address" near the bottom of the page).
Every Kindle has an email address. You can find yours by going to your Personal Documents Settings on Amazon. It will be something like "email@example.com" or "firstname.lastname@example.org". Enter the "you_123" part in the form above and click the Save Kindle Settings button.
Second, Go to Your Leanpub Library and either Send to Kindle or Download
Once you're set up, go to your Leanpub library, select the book, and click the "Send to Kindle" link you will find there. (If you don't see the "Send to Kindle" link, that means this book is not available in the MOBI format required by Kindle.)
Alternatively, you can download the MOBI file for the book onto your computer or device and use one of Amazon's apps to get a book onto your Kindle.
Third, Go to the Docs (not Books) Section on Your Kindle
[UPDATE October 28, 2019: A customer has informed us his books showed up in the "Downloaded" space in Kindle, and he didn't have to go to a separate "Docs" or "Documents" section, so this part of our guide may (we hope!) be outdated.]
To read your Leanpub book on your Kindle, you will have to go to the "Docs" section underneath All Items in the main Kindle menu. Send to Kindle puts our books in "Docs", instead of the "Books" section. Presumably, Amazon must only think something is a book if they sell it!
A Note About File Sizes
The "Send to Kindle" feature on the Leanpub website only works for books whose MOBI versions are under 49 MB in size. If the book is over 49 MB in size, we won't show you any Send to Kindle links for it on the Leanpub website or in Leanpub emails, since they won't work.
Alternatively, if you have the Kindle app installed on your computer, once you have downloaded the .mobi file from Leanpub, then if you double-click the file it should open in the Kindle app. Next, connect your Kindle to your computer via USB and manually copy or drag the .mobi file for the book from your computer into the "Documents" folder on your Kindle.
To learn how to get a file larger than 50 MB into the Kindle app on your iPhone or iPad, please read this blog post.