Just off hand it doesn't sound like the key is really turning as it should. There should be some resistance as you turn it and it should drop back into the second position when released. Does that happen? Can't tell from the video. Also, what do you hear under the hood when you turn the key to the start position? If the key switch is working, you should hear clicks (relays) as electricity is directed toward the stater. The obvious failure points are the switch, the solenoid, the starter motor, the starter motor itself, the wires that connect them and the connections, and maybe a fuse too. From under the hood, can you hear the solenoid click, does the starter motor turn or make any noise at all. For an automatic there are also the connections at Park and Neutral that allow the vehicle to start.
