I recommend you get a game console - all three current and last generation consoles support Amazon video (Wii/Wii U/PS3/PS4/Xbox 360/Xbox One). Plus they can play games.
There's a pretty big difference between getting a $35 Chromecast vs even last generation's console. Other than the cost, I also feel that now that I'm a parent, my consoles are just terrible reminders that I will never again have the time or inclination to play any of the wonderful games that go with them.
Or a cheaper option of a blue ray player (I have a six year old sony blue ray player which has the Amazon video app )which you can probably get now for under $50.
And 37 menu navigation options later (including the always fun 'spell words like you're texting in 1997' search functionality) you can finally watch something.
As opposed to the 'use the native UI on your phone, then hit one button to cast it' experience.
We are Amazon Prime subscribers, but I can't remember the last time we watched anything on Amazon Prime Instant Video... it's just too much of a hassle...
This is definitely why I prefer the Chromecast experience over loading Hulu through the PS3 (which is actually also pretty good, given that Daniel Tiger is always on the front page). It's a small difference, but when trying to get myself and my daughter ready in the morning, any reduction in friction is appreciated.