Narrow bandpass filter using maximally flat LPF prototype design
'Andy aka' has stated the best answer. Adding any kind of filter to the final driver stage will be pointless, as the coil and cap values are chosen to be resonant at 13.56kHz. If these L and C values are resonant at 13.56kHz, then they will oscillate at exactly 13.56kHz (ignoring parasitics) and thus, give you the most optimal power transfer.


  The Q, quality factor, is a measure of the “goodness” or quality of a resonant circuit. A higher value for this figure of merit corresponds to a more narrow bandwith, which is desirable in many applications. More formally, Q is the ratio of power stored to power dissipated in the circuit reactance and resistance.


So in other words, if you want a high Q for the transmitter, use:


High-voltage, low-ESR (and low inductance), quality capacitors (NP0, X7R, etc.)
Low-resistance, low-capacitance wire in the coil (Litz wire?)
Good PCB layout design (low parasitic effects.)


You may find that driving the output stage via a squarewave at 13.56kHz will yield more output power, as doing so creates faster transients (imparts more harmonics) into the coupled magnetic field. But it does so at a higher input power, so thus is not most optimal both in terms of power usage and signal purity. The whole point of parallel resonance is, that as little energy as possible is added to the system to keep it oscillating.

I think we are scratching our head at why you want very pure-frequency transmit energy. If you want the receiver to be very selective and only allow this particular transmit frequency, use the 13.56kHz bandpass filter on the receiver end. Note that -3dB is half the power, -6dB is a quarter, etc. How little power does the receiver need to function? That will allow you to estimate a frequency response:


The coil distance (in air?)
The magnetic coupling efficiency (a core material other than air?)
"Q" or steepness of the transmitter
"Q" (order) of the receiver and filter network
The minimum required receiver power to function


This all said, I'd consider a frequency higher than 13kHz. That is within the range of human hearing, and microscopic movement of the components (the opposite of microphonics) is likely, leading to us physically hear the high-pitched 13.56kHz tone. (Human hearing has a sensitivity of 20 micropascals!) 25kHz would be better for most of the population.