The Amazon Fire HDX8.9 tablet has had Miracast problems working with various devices such as Amazon Fire TV and Roku 3, and unfortunately has the same problems with the Roku 4.
On the Roku 4, first go to Settings and make sure Screen mirroring is enabled.
Roku's implementation of Miracast requires that both the Roku and the devices you are trying to wirelessly display be on the same WI-FI network.
To start Display Mirroring, you need to go into the Fire HDX 8.9's Display & Sounds in Settings. Select the Roku 4 to start the connection process.
Unfortunately, the Roku 4 remains frozen with the screen mirroring (beta) screen and the Amazon Fire HDX8.9 tablet is unable to connect.