Sorry to hear that you having an issue and we are more than happy to assist you with this. I have attached a link below that I hope will help and clear up any questions or concerns that you may have.
Hope this helps,
Sling Moderation Team
but the link you gave, says that "SlingPlayer must be running in High Quality (HQ) streaming mode" which is a mode that is unwatchable due to constant buffering of the video (i guess due to end2end bandwidth constraints in my situation).
Also, I am not prepared to buy a cable(!) solution for $39 for something that the already expensive sling player app should support.
Summary: Airplay does work in HQ mode (unfortunately, showing the same unwatchable video on the TV in my situation), but not in SQ mode.
Clearly it is a bug in SQ mode airplay, so please tell me at least that this bug in the SlingPlayer is put on the roadmap for a fix.