Need a workaround for no inline video play with iOS (iPhone)

On the website I am working on, understood.org, we have a page where a video plays, then upon ending it starts a javascript game, and then after that game is complete plays another video. This works on all platforms pretty well except for the iPhone. The problem is that on the iPhone it plays the video in a full screen native player it seems, and so the trigger to start the simulation and close the video doesn't work, and instead it skips directly to the next video. This could all be fixed if I could just tell Safari to play the video inline, but I have looked around and seen on numerous questions and forums that this is just not possible. Any ideas? Do we need to use a different video player or something? The example I am working with can be found here http://ift.tt/1I1bECG