"Antenna matching networks for dummies"?
The frequency is low enough for measurements with oscilloscope. Add a small few ohms resistor in series with your antenna. Be sure that the resistor is not a coil. It must be resistive at the operating frequency.

Feed with a signal generator a sinewave into the antenna at the right operating frequency. Measure at the same time the voltage and the current and their phase difference. You can calculate the impedance of the antenna as a complex phasor. A single frequency LC matching network for transforming the measured complex impedance to a wanted resistance is a popular training problem in phasor calculus books.

You can also tune the system when measuring the antenna through a matching network. There are plenty of variable tuning networks available in sourcebooks. The tuning is quite a job without knowing anything beforehand. The goal = no phase shift between the current and voltage, the resistance = the specified one.

WARNINGS:


all reflective parts near the antenna change its behaviour radically. The final placement would be optimal. When doing the measurements you can test how sensitive your system actually is.
the cable between the antenna and the matching network should be as short as possible to avoid long reflections in the cable.
the cable is a part of the reactive system, so it cannot be changed between the measuring point and the antenna, the calculated matching network is right only for just the point where the measurements were done.


ADD: This is not beginner's job. One must understand and be able to perform complex impedance calculations, if he wants to calculate the network. Also the no-nonsense measurements with oscilloscope and signal generator even at so low as 13,56MHz are far from trivial. Get an experienced local partner. By trial and error you will get to nothing except burn your transmitter because a bad mismatch can cause a serious overvoltage to output.

ADD2: If you happen to understand well the math (=impedance as phasor) behind matching but feel yourself too unsure to perform the calculations reliably, you can use a circuit simulator to find the right component values for the network after good enough guessing. Reliably measured complex impedance is a must for this as the starting point.

ADD3: I met the following link that contains plenty of probably interesting material: https://learn.adafruit.com/adafruit-pn532-rfid-nfc/downloads

There's also a forward link to antenna design &amp; tuning quide. It contains material much more focused to thislike systems. My own story is only a generic one which holds nothing specific for questioner's  actual system. For example there is suggested to use 2 identical antennas and tune for max. power transmission. Also there's described the right measuring probes that are up to the task, if one uses an oscilloscope and much more. The quidance is clearly based on actual experience, not on the theory and general measurement knowledge like my story.