I have successfully rooted my old Kindle Fire HD 7" tablet and installed Android 4.4.2!
I noticed I kept getting a “Google Keyboard has stopped unexpectedly”. So how did I get that fixed? By empying the cache of the Google Keyboard app.
Now my Kindle Fire HD 7" runs Android 4.4.2 with no problem!
And the best part? I can still install Kindle from the Google Play Store and have access to my Kindle library.