Once you started actually talking about your problem, it became clear that you were indeed asking an xy question. The question you really want answered, is "How can I identify a drone by blinking its orientation lights?" or even better, "I'd like to identify a drone by blinking its orientation lights. What should I be thinking about?". Although the latter is a bit vague, I agree.
With that in mind, it's clear that blinking the orientation lights at 1 to 5 kHz will not do you any good. If you're looking at it with your eyes, kHz is way beyond visual fusion rate, so there's no way your eye can distinguish between drones with different flash rates. If you're looking at it with a camera, 1 to 5 kHz is far beyond the frame rate you'll get with any reasonable camera. To identify a 5 kHz flash rate from camera footage requires a minimum of 10,000 frames per second, and that is hardly standard kit.
All things considered, it would seem your best bet is to point a telescope at it with a very high-sensitivity detector and a frequency detection circuit. Such a detector would require a very narrow bandpass filter at the LED wavelength.
So, rather than insisting that what you want to know is how to flash an LED, you need to start thinking about just exactly what you want to do from a system viewpoint, and determine if you CAN identify a drone by flashing its LEDs. Only after you've done your system calculations should you start worrying about how to do it.