I have an Amazon Fire Stick which I mostly use for Amazon Prime Music.
I have both generations of the Amazon Fire TV and love them... I have Broadband 35 down with 5 up and have no problem with 1080p streams...
Have you tried streaming 1080p movies... I am curious as to whether they have enough cpu to do it too.
I just want to make sure we are on the same page... I have the Fire TV Streaming Media Player and not the Fire TV Stick...https://www.amazon.com/Amazon-Fire-TV-Streaming-Media-Player/dp/B00U3FPN4U/ref=sr_1_3?ie=UTF8&qid=1478973285&sr=8-3&keywords=firestick
Fire Stick and Fire TV with Kodi side loaded. By far the best thing I have used with Kodi. Everything else has lagged way to much to use.