I have noticed that sometimes I cannot watch my Amazon Prime videos on a rooted kindle fire. I then noticed that with two apps, I could watch these movies, sometimes, but then sometimes it just wouldn't work, so here I will detail the two apps you will need and the exact steps to easily without Amazon Prime Videos on your Kindle Fire.
You will need to grab the following file from google play or search the web for the apk file. The program is called:
Voodoo OTA RootKeeper
You also have another program called Superuser which was installed when you rooted.
Now, go to your Amazon Video and try to watch one of the prime videos. You will notice your "Watch now" button is greyed out and you cannot click it to watch your Amazon prime videos.
Now first thing you will need to do is tap on the little gear in the upper left hand corner. Once you tap that, a menu will come up. Select More.
Once you tap on More, you will be at the settings screen.
Now tap Applications. By default it shows you your running applications, which is exactly what we want. Now tap on Amazon Video. Once that comes up, tap on Clear Data under the storage section.
Now once you have done that go back to your home screen and launch the Voodoo OTA Root Keeper. There you will want to click on Protect root, once you tap that tap on Temp Unroot "Keeps Backup"
Now go back to your home screen and launch the Amazon Videos.
You will now see you can watch your Prime or purchased videos without any issue. Now, here is the thing you can watch all you want. You may even go back and restore root in Voodoo OTA Root Keeper. Then continue watching your videos.
Everything will be fine, unless you stop the Amazon Video application, or reboot your Kindle Fire. If you do this, just go back into Voodoo OTA Root Keeper and temp unroot again, but make sure to follow these steps again, as you must kill the amazon video application before you remove root, as it does a root check on launch.