http://www.amazon.com/Amazon-com-Amazon-Video/dp/B00N28818A

I found that for me by going to http://smile.amazon.com/Amazon-com-Prime-Instant-Video/dp/B00N28818A/ref=sr_1_1?ie=UTF8&qid=1412286230&sr=8-1&keywords=instant+video it showed that Amazon Prime Instant Video was not compatible with any of my devices (I have a 2.3 phone). This is probably why it wasn't enabled in the app.

http://smile.amazon.com/Amazon-com-Prime-Instant-Video/dp/B00N28818A/ref=sr_1_1?ie=UTF8&qid=1412286230&sr=8-1&keywords=instant+video

There's an app for that! Instant Video on Android, that is.

http://www.amazon.com/Amazon-com-Prime-Instant-Video/dp/B00N28818A

Here it is.

I notice it's quite buried when searching for some reason. Make sure you have updated your Amazon app too