First, if you don't have it already install the Airplay app on the Roku device. If your Roku TV or Roku device is still not showing up in Airplay or screen mirroring options, try this. Roku 10.5 software caused an issue with Airplay but can be worked around as follows:
On your Roku device home screen, go to Settings>System>Power>System Restart>Restart. After it finishes, (the TV will turn off, then on again and take 30 seconds or so to get back to the home screen) proceed as follows.
From the home screen on the Roku device go to Settings>Apple Airplay and HomeKit and select. It will reload the Airplay app (it took about 20 seconds on my Roku TV). Now turn on apple Airplay (it is defaulted off with the reset). Now, when you go back to your apple device, the Roku device should show up as a screen mirroring or Airplay choice.