If you have not solved your problem with how to get it on the TV, there are only a few choices. Amazon are kinda being a-holes about streaming platforms, so this is what you have:
Amazon app built in a TV
Roku
Amazon Fire TV
That is it to get it on the TV, there are of course hacks, but that is the official ways.
I use a newer roku personally, works great.
The amazon app on IOS does not support Chromecast. There is a non Google Play Amazon app for android, but it sucks and again no Chromecast.
This all has to do with I assume Amazon getting you to buy episodes on your amazon account, and not itunes or google play store. Not sure, other than legacy, how it ended up on IOS anyways. Roku doesn't sell content, so they are OK with that.