That is the learning experience so no need for self criticism. Chapter 4 in this document https://ww1.microchip.com/downloads/en/Appnotes/Atmel-42183-Design-a-Buck-Converter-with-XMEGA-E_AP-Note_AT04204.pdf shows a high frequency mosfet driver (for a P channel mosfet) which is made of discrete components and has an explanation of the circuit. Bear in mind, however, that is high frequency application using a 32MHz MCU. You are talking about a frequency only 20kHz and, as has been pointed out, is unlikely even to require a special driver.
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