I guess the transformer is your stand-in model for two coils close together. Coupling coefficient is a way to simulate the effect of weakening magnetic flux over a distance. How to select coupling coefficient? This is hard to answer. 99%? 90%?
Your schematic appears to provide pulsed DC through your sending coil. This allows merely 50 percent utilization of the coil. Efficiency goes up when you cause it to emanate maximum power. Therefore send bipolar AC through it. A simple self-oscillating circuit is sufficient. Start with a lesser supply voltage than 12V.
Your receiver section needs to be tuned to the exact frequency beaming from the sending coil.
Your output section consists of several components, requiring that your receiving coil produce several volts. I believe that 2 led's anti-parallel by themselves can be powered from your coil.
Say a red led needs 2VDC 10mA. That's .02 W. Suppose you feed 1W to your sending coil.The led's should illuminate if you manage to obtain just 2% efficiency.