Download and install it. It should work; however, it seems as though the app gives people problems across multiple platforms. Some reviews on Amazon even say that users have 2 Kindles (of the same model) and it will work with one but not the other.
If you're Kindle won't allow you to download and install it through the Amazon app, then download the *.APK file using another device (or the browser) and install it manually. You won't know if it truly works for you until you try it.