you can make a variable capacitor with a variac where the variable ratio changes the capacitance according to the ratio squared.
but this won't work if the circuit sets up a dc voltage across the capacitor.
in anycase, a 15uF cap for a small 1 amp motor could easilly fit on a pcb, such a capacitor would be about an inch in diameter and two inches long.
I don't know why they have a 50uH inductor in series with the capacitor.
btw, this is not an lc resonant application, line frequency is no where near 5.6Khz
as far as finding the capacitor value, its not critical. just set it up so the current flowing through the cap is about the same as the current flowing in the other coil, at full load.
i have a few motors setup like this, one of them uses a 30 or 45 uF motor run capacitor(maybe 1/12hp), the other uses two motor start capacitors in series, each is about 200uf iirc. (chair lift motor, intermittent duty, about 1/8th hp)