If your a fan of the Eaa Video player content most of it can not be found on youtube. so for a Smart TV or Smart Box in the work shop this makes watching the content difficult.
Free App called Video & TV Cast for Amazon Fire TV, it will need to be installed on both your Fire TV along with a IOS device (iPhone, Pad)
Fire TV Link:
http://www.amazon.com/dp/B00V4V3TWA/ref=pe_385040_117923520_TE_M1T1DP
IOS Link:
https://itunes.apple.com/us/app/video-tv-cast-for-amazon-fire/id979567301?mt=8
This is not an Air Play app, you do need the apple device to get to http://eaavideo.org/ with in the App, there is a cast to notification after that you can turn off the Apple device and it still plays on the Fire TV.
once all installed open the app on the Fire TV
in the App on your IOS device Search for Eaa Video
Click the upper Right Icon to bring up the devise selection
Click on the devise
find the video you want to play and get it started on the IOS
The video will start to play
Once the prompt comes up Press the Cast now button to send to the Fire TV, you may need to press the upper right icon to get it to prompt
All done, at this point the phone can be turned off and the Fire TV will keep playing the video.