I get all my .apk's from my smart phone.
*** Note: My smart phone and my Kindle HD 8.9 are both NOT rooted. ***
This is what I do:
1) Download and install the app of interest onto my smart phone.
2) Backup my smart phone installed app using Astro File Manager
3) Go to the /Backup directory of my smart phone and copy the apps *.apk to Dropbox
4) Open up Dropbox on my Kindle Fire HD 8.9
5) Copy and Install the app *.apk from Dropbox to my Kindle Fire HD 8.9
Note that there are several ways to transfer an *.apk to your Kindle once you have it on your smart phone. You can use your usb cable to pass it from phone to computer to kindle, you can email it, or you can even use WI-FI File Explorer which is an app that turns your Kindle into a computer harddrive through your Wi-Fi connection. It is just that I have found that passing it to Dropbox is very simple, fast, and also gives me a way of backing up the app as a nice side-effect.