Don't know much about how the chromecast works with the Kindle Fire although I think some amazon based stuff is 'deliberately incompatible with other stuff (e.g. can't play Lovefilm on android devices apart from Kindle Fire).
I have a chromecast myself running on my android phone, tablet, laptop etc and it's a great gadget. Apart from watching netflix wirelessly, you can 'cast' youtube videos to your TV as well as watching on demand TV from BBC, Channel4, etc. You can also cast live TV to your TV (sounds silly I know but it saves me paying for a second cable box in my bedroom) via a free android app called FilmonTV.
I'm actually thinking of getting rid of my TV package, land line (can't remember the last time I used my landline-can't even remember the number) and going internet only...I'll still need my TV licence but will save about £50 a month that I can buy bushcrafty things with instead...