16,000,000 cycles/second/24 samples means absolute best frquency possible would be 666,667 Hz
divide by at least 6 if all the instructions in reply #2 executed in 1 clock cycle
down to 111,111 Hz
each SPI transfer is 8 clock cycles, so divide by 16
down to 6944 Hz.
So 20 KHz does seem like a stretch.
Let us know how fast you do manage tho.