For swotching applications, beta has no meaning: you should provide sufficient Ib to saturate the switch. Usually Ib is at least 1/10 that of Ic.
In your case, Ic is determined by that resistor (together with supply + the led's forward voltage drop).
So the calculationb is backwards: select your resistor to get the desired Ic and from there, get the required Ib and calculate the corresponding base resistor.