This may be a little too late for you now but I have been able to fix my issue.
The short version, I created an el Capitan boot disk on an 8GB USB drive using the free app Disk Maker, downloaded the latest El Capitan from the App Store (10.11.5) and did a clean install of El Capitan. Once I rebooted my PC, Airplay was working properly.
Unfortunately, I then lost Airplay functionality again when I reinstalled my backed up Apps using time machine. After a bit of investigation, it turns our that Airparrot was the cause of my problem. Although I had uninstalled the application some months ago there were still some driver files in the system.
I manually deleted the following files at the Terminal command line using the sudo command,
APExtFramebuffer located at:System/Library/Extensions
AirParrotDriver located at:System/Library/Extensions AND /Library/Extensions
then emptied the trash to be on the safe side and did a system restart.Once I'd done that, it was back to normal - 8 months later!
I thought I'd mention the second bit because, apparently there are other third part apps that have been causing the same issue. e.g.. Air Display, Splash Screen, so it might be possible that you have an uninstalled app that still has it's driver files sitting in the system causing the same type of problem?
If you're comfortable with the command line, this would be the quickest solution.
The more time consuming option is obviously to do a clean install and manually install all the relevant applications that you are still using. Obviously, by reinstalling the applications using Time Machine is a lot easier but of course, you run the risk of reinstalling data from a third party application that may have been the cause of the problem.
Anyway, I hope that you have already sorted your issue, whatever the cause may have been. If not, then maybe this will be of some assistance. Good luck.