Have you tried using a pot to vary the resistance to see if you can find a "sweet spot" for the resistance?
I am trying to have a mathematical basis for choosing values,
Ouch! $6. Not much point in getting a stinking fast MOSFET if the circuit driving it isn't going to switch it faster than 5uS anyway.
BTW, that rule of thumb of base current being 1/10th collector current is in the datasheet for the 2N2222 and 2N3904. It isn't that the gain is 10.