I have a problem with choosing most appropriate transistor to control backlight of my LCD display. According to datasheet its \$I_f = 60mA\$, and \$V_f = 3.2V\$. My circuitry is all 3.3V based. I want to supply PWM signal from my STM32 F411CEU6, but it can only supply max of 25mA per GPIO pin. Due to rather large \$V_f\$ there is not a lot of space left for Current limiting resistor, and \$V_{DS}\$ or \$V_{CE}\$. I have found 2SC5585TL NPN BJT (datasheet). According to it, if I were to supply 6mA at Base (\$\frac{I_C}{I_B} = \frac{60mA}{6mA} = 10\$) \$V_{CE}\$ should be around 25mV, which leaves 75mV for current limiting resistor.
simulate this circuit – Schematic created using CircuitLab
Will this work as intended? Are my calculations correct? What is the preferred transistor type here BJT, or MOSFET?