To me the quality of sending it via Airplay isn't all that good.
Looks like Apple has a pretty pricey adapter cable that can connect an iPad to an HDMI connection. Or maybe I should also experiment with connecting my kid's Kindle Fire HD to the TV to see how the Amazon Prime quality is on the TV.
I have used the Airplay feature built into the Amazon Instant Streaming app on my iPhone, and that quality has been pretty decent. Make sure you are using the Airplay feature of the Amazon app itself, not the "Airplay Mirroring" feature that sends everything on your iphone/ipad to the screen. The latter won't look good on a big 1080p tv, since it is just matching the resolution of your iDevice. The airplay built-in to the Amazon app (I think) essentially sends your Apple TV a link to the appropriate Amazon server, so it can stream directly at the appropriate resolution based on your Apple TV's internet bandwidth.
Frankly, if the Airplay (using Amazon app) quality isn't good enough for you, it is likely not Airplay itself that is the bottleneck. It may be your internet connection or Amazon's servers, and hence using a hardwired HDMI cable likely wouldn't make much of a difference.
I wouldn't bother with the Apple HDMI cable. I don't have that particular cable, but I have had the Apple Video (RCA) cables for previous generation of iPad iPhone (i.e.: the 30-pin connector to RCA). Generally speaking, those cables aren't as versatile as Airplay because they don't tend to support most apps on your iDevice. Usually it's just the "Videos", "Photos", and "YouTube" apps. Maybe Netflix as well (don't remember). I suppose you could buy and then return if it doesn't work, but I wouldn't hang my hopes on it.
If you are going to be using Amazon Instant Streaming a lot, it's probably worth getting a Roku or a Blu-ray player that has it built-in, just for the convenience of not having to Airplay. I recommend the latter, as that gives you the added capability to play BD content, which the Roku does not.
But, again, you may find that the picture quality is no better than using Airplay. I have a 2013 model Sony Blu-ray player that has Amazon Instant Streaming built in, and the video quality is good. But using my iPhone and Airplay yields the same video quality as far as I can tell.