Unfortunately, amazon instant video is not an option on your tv.
This set does not show the app or have an activation code because that functionality is not supported on this model.
Your tv, and virtually all other 2012 and prior models, do not support amazon content any longer. The marketing agreement for those products has expired, as has firmware support.
Your tv is actually well past its designed service life at this point, and for that reason Samsung is not supporting it. To get that functionality, samsung would have to renew a licensing agreement with amazon as well as publish a special software update for your specific model, which simply won't happen at this point, they are more focused on selling newer tvs. Your other apps will stop working in the near future as well, if they haven't already. Youtube and skype were both dropped recently for example.
So the tv can no longer use amazon directly, leaving you with the options of either replacing the tv, or connecting up an external streaming device, which is what I would suggest, such as a roku or fire tv and get all the same functions and many more, with better performance and lower cost, and better long term support.
My apologies for the news,