The latest Kindle Fire HD is not officially supported by Flash Player. Please see this page for a list of all certified devices:
That said, some users have been able to get Flash Player working on the Kindle Fire by using Dolphin, a third party browser.
The first gen kindle fire comes preinstalled with Flash, so you can just toggle the settings. But all other gens require installing flash and the dolphin browser. Check out this site for more info: http://kindlefiretuts.com/how-to-install-flash-to-the-kindle-fire-all-gens/
From Amazon Kindle Support (07/31/2013):
"The Amazon Silk web browser on Kindle Fire 1st Generation supports some, but not all, versions of Flash. Adobe Flash is pre-installed on a 1st Generation Kindle Fire and is disabled by default.
If you see a message to install Flash while using your Kindle Fire, this means that the content you're attempting to view uses Flash. If you want to view the content, you'll need to enable Flash through the Silk settings menu.
If a web page suffers from slow performance due to Flash, or you're unable to view a web page with Flash enabled, we recommend switching to mobile view when browsing a Flash-heavy website to see if that solves your problem:
Some popular websites may also have an app available. Check Amazon Appstore, accessible through the Apps library on your Kindle Fire, for availability.
The Amazon Silk web browser on Kindle Fire 2nd Generation, Kindle Fire HD 7", and Kindle Fire HD 8.9" doesn't support Flash.
Some sites use alternative web technologies to Flash in their mobile versions. We recommend switching to mobile view when browsing a website with Flash content to see if that solves your problem:
Some popular websites may also have an app available. Check Amazon Appstore, accessible through the Apps or Games library on your Kindle Fire, for availability."
Android dropped support of Flash BEFORE Adobe stopped development of the Player that is no longer supported. Android sucks too I guess, but I don't have a kindle, so I reallly can't say.