What version of the Kindle reading app do you have? (In the Settings app go to: Apps & Games -> Manage All Applications -> swipe left multiple times to the "All" tab -> scroll down to "Amazon Kindle". Select this app to show the version number.)
Fire OS 5.3.1.0 and 5.3.1.1 come with Amazon Kindle version 10.11_1197791110. The latest version is 10.16.62_1198819010, which corresponds to Kindle for Android version 7.4.
Another HD6 user reported a similar problem in
this thread. I don't know whether the problem was ever resolved.
I do know that Amazon requires a secure connection to download component updates. If your Wi-Fi connects via a proxy you may have problems.