I can't seem to view flash content on IE Windows Apps in Windows 8.1; however, it works if you view it in desktop mode. I used all the of the troubleshooting steps in this article, and as mentioned, it does indeed work in IE desktop, but it still shows the error message in the app.
Error Message: Oops! You don't appear to have Adobe Flash Player installed. Please install flash player to continue or try one of our mobile apps.
Example: the Digitally Imported app, available from the Windows App store.