I made a similar mistake when I was getting an Xbox controller that would work with my computer, but as imeem
said you need the wireless receiver, because the play and charge kit isn't meant for sending signals for games through the cable. The cable is meant only for charging the controller, this is the same when its connected to the Xbox. Even though its plugged in the wireless controller still only transmits data wirelessly
, the cable only charges the controller. I believe that's why it doesn't work like that.