Oops, I shouldn't say Qgs, that's a separate part of Qg(tot).
Ceff = Qg(tot) / Vgs(on)
So if it's 5.5nC, then Ceff is 550pF. Give or take that you're running it at 12V, so Qg(tot) is probably higher, but Ceff may or may not be the same. Anyway, it won't be grossly different.
Actually, maybe it's not 12V gate drive, there's an emitter resistor and presumably a logic level input. (104 ohms, that's really oddball?) What logic level? CMOS, 3.3V, 5V? TTL? The collector only pulls down to a bit above the emitter voltage, and if the emitter voltage is set by the input voltage, it might saturate at say 2-4V depending on source. Which puts 8-10V across the MOSFET, which is still just fine.
At turn-on, you have a fairly strong current from the BJT. If it's about 3V over 100 ohms, that's 30mA (minus up to 1mA into the pullup resistor, who cares), and we can expect the gate to turn on in about
t_on = C dV / I = (550pF) (10V) / (30mA) = 0.18us
which is fine.
Turn-off is the bigger concern. Here you have no current from the transistor, only the resistor discharging it. We draw the equivalent RC circuit, 550pF in parallel with 10k, initially charged to ~10V. This has an RC time constant of 5.5us, which is quite long, quite a bit longer than the turn-on case. A smaller resistance may be desirable (say 2.2k or 1k). A better drive circuit isn't really necessary, but is an option.
5kHz is 200us per cycle, so you should want switching times much less than that, say, below 2us.
Finally, protection:
- The motor is an inductive load, right? (Possibly a BLDC motor isn't, I'm not sure; a brushed or universal motor definitely is.) When the transistor turns off, its inductance will act to continue the current flow, even as the motor voltage drops to zero, and below (negative). Typically, a diode from GND to output is used to clamp this. The current decays through the diode, and voltage is limited. Simple and effective. (The diode, transistor and a supply bypass capacitor should all be placed nearby.)
- What is the motor's maximum current (LRA)? You should want some kind of protection here, either brute force (the FET is beefy enough to supply the excess current without blowing up), or active (sense current flow, and turn off the FET if it gets too high). (Fusing is a possibility, but not practical here: transistors take ~100 microseconds to blow up, while fuses take ~10,000 microseconds to blow up. Definitely recommended for protecting the wiring, but it won't save the transistor.)
- What is the power source? If this is automotive power, mind that the supply can be quite messy. I wouldn't recommend taking gate voltage directly from it. This can be fixed easily by putting a zener diode from gate to source, say 1N5245 or equivalent. If the supply voltage jumps up suddenly, the diode clamps Vgs, and the remainder is dropped across the BJT -- which since it's limited to 40mA or so, is fine, it won't explode (at least, not instantly).
- Also if Vds(max) is pretty low, that's a concern. I wouldn't recommend less than 30V rating for automotive; >= 60V is industry standard.
Tim