The new app is designed for phones, although it is working on some tablets. It is a somewhat complicated install procedure because you first must install the Amazon Shopping app, then download the one for Instant Video.
I was able to install the app on my Samsung Note 2 with no problem. I could not, however, get it to work on my Samsung Note 8.
Both Teleread and The Digital Read have posted workarounds for the flash video problem in the past. Today, the MobileRead forum led me to another install method detailed on the Droid Life site. This install involves directly downloading the Instant Video .apk file from Amazon.
I tried the Droid Life method and it installed correctly on my Samsung Note 8 with no problem. 🙂
Hopefully, one of the methods will enable you to get the app working correctly on your device.