This is a general issue with Kodi, Airplay seems only compatible up to iOS 8.
It's not an issue with Kodi. It's an issue with Apple (specifically IOS 9) always changing things. At the moment, there is nothing to fix beside the devs will get it to work (until the next Apple update when it will fail again).
It's not an issue with Kodi.
Sorry, this IS an issue with Kodi, as other software is able to simulate the Airplay protokoll. Even for Kodi seems to be an (commercial) addon, which is Airplay compatible:
LibreELEC
EDIT: just tested with the actual release: deactivate the last option in the Airplay settings (video and picture support) and it plays at last music with iOS 9.x
So then ask yourself why you have to pay for that add-on (from which we neither have a link nor a name atm). Then think about "FOSS" (free and open source software) and what that means. My guess is, that the devs of that mystic add-on are paying apple probably to get the informations Kodi doesn't get. But I might be wrong.
In general, Kodi as a FOSS, is NOT the issue if you have to pay for a service to get the needed informations how to use that protocoll. It's fine, if one is willing to pay for a service. But don't blame Kodi not to pay for a service to satisfy such a company to use their possibilities. Otherwise you might have to pay for Kodi as well....and that wouldn't be the thought which is behind FOSS.
Noone blames Kodi under Linux not being possible to play native BR discs because of missing licenses. Luckily there is a way around that. And that way is for free, too. But everyone was fine to accept, that Linux has to be a free OS and therefore missing the option Windows provides instead.
So still not a Kodi 'issue' in my point of view. But maybe our point views differ in a bit and that's fine 
Actualy, Airplay audio works fine. You need to activate "expert" settings (not advanced) and then turn the option "Enable Airplay "Videos" and "Pictures" support" off under services/airplay. This option only works for iOS8.
Then you should turn that option off. It works with iOS9 (I'm on 9.3.1), but only if you turn this option off. This is displayed under the option when highlighting it as well. Bad default value if you ask me but that's how it is.
So still not a Kodi 'issue' in my point of view. But maybe our point views differ in a bit and that's fine
I'm sorry, if I used "issue" in a wrong way, as english is not my native tongue. I meant, Kodi can't do it, for which reason ever. This should not "blame" Kodi for anything. Please don't think I'm ungrateful, I'm using Kodi since his original destination on the Xbox as XBMC and think this is the best media-software in any way.
In my second post I edited, that the actuall Kodi indeed can play Airplay audio again with iOS 9.x, when disabling the video and picture option (I'm not using the english version, so I can't name it correctly), which is a great deal for me and I'm very happy, that this works again.
iPhone is a mobile phone developed by Apple. siri not working iPhone combines an iPod music/video player, a mobile phone, an internet web and email client, and a handheld application platform into one device. iPhone is intended to compete in the smartphone market.Apple has released ten generations of iPhone models, each accompanied by one of the ten major releases of the iOS operating system.
