The full version of the Calm app is not currently available on Kindle or Amazon devices. Amazon devices only allow updates from the Amazon store which unfortunately does not have a recent version of the Calm app and it does not support subscriptions. While we do hope to have the version in the Amazon store eventually, at this time the subscription version is only available for Amazon devices that can download apps from the Google Play store.
If your Kindle includes a web browser, you may be able to access Calm's content through your tablet's mobile browser. We recommend trying to open www.calm.com/meditate and see if you are able to log in and access the meditations there. The site is not optimized for this, but some users have been able to successfully use their tablet browser for Calm.