I'm planning to do a turn-on-back-light construction with transistors like this:
Where the LED is the back light of the LCD ILI9341.
But to select the resistors, I need to find first how much current the back light of ILI9341 take. Do you know how much it want?
You are asking a trick question.
The assembled LCD module consists of a display module assembly with a suitable backlight assembly.
The fact that it has a ILI9341 TFT controller has very little to do with the backlight, even though the brightness may be controlled via ILI9341.
The answer to the backlight specs would be in the datasheet of the whole LCD module, not in ILI9341 TFT controller datasheet.
So you must know the part number for the whole module to find the datasheet for it.
There are a bunch of similar products. Most seem to have a small resistor of several ohms in series with the LED supply to allow operation from 3.3V, and perhaps 4 white LEDs in parallel. You may be able to see the resistor on the back of the PCB.
In any case, a typical recommended current is 60mA. You can probably count on it being less than 100mA. If you are using a PNP transistor such as 2N4403 or 8550 the base resistor should be something like 510\$\Omega\$. 10K is okay for the NPN base resistor.
Just as an FYI, you are looking at several components.
The ILI9341 is the controller chip (only) made by Ilitek
There is an LCD panel that includes the controller chip. That's the part with the FPC (flat cable) coming off it. Another manufacturer makes that. It has only 4x LEDs on it, with no current limiting for the backlight- and usually allows for 4 resistors. Volume OEMs would buy that and attach the FPC to their bespoke PCB.
Finally there is the red colour PCB. A third manufacturer likely makes that. It is essentially a "breakout" board for the panel that makes it easy for hobbyists to use it, and has a very few parts on it (the aforementioned dropping resistor, perhaps a touch controller chip and maybe an SD socket regulator etc.). The supplier should be able to give you a schematic (maybe like that one) for it (but it won't help much on its own because you also need to know the panel characteristics).
Your LED display has nothing to limit its current. But since the 10k base resistor R2 on the driver transistor has a resistance too high then the display will be VERY dim anyway.
