Acannell observed (roughtly) that "95% of forum denizens tell you you are wrong. 5% actually get enthusiastic and help."
Where do I stand?

Well, I still say an MCU is the way to go. Indeed, a problem like this got me into microcontrollers in the first place. I wanted a circuit to control the brightness of the light on my mountainbike, with buttons for brighter, dimmer, max and off. The logic chips were getting hairy but a simple PIC12C508 (yes, a C version, I've still got a couple of windowed ones around 15+ years later) solved it and even added more features like flash mode and an illuminated switch lever when the light was off.

However, to put my money where my mouth is, I cobbled a circuit together using 75LS191 chips. I fired up CircuitMaker (you can download a student version here) and bodged in the attached circuit. It basically works although it needs cleaning around the edges, such as proper counter setup when the reset signal arrives.

(Hopefully obvious but the blue wires are logic 0 and the red ones are logic 1. :-)

Top left is a clock generator. The switch banks on the left set the input values. The semi-connected resistors on the right are just so there is a wire connected to various pins in order to see the state of said pins. The NAND gate top right should be an AND gate instead, the output of which is the N x K pulses.

The upper 'LS191 continuously counts down from the load value N and resets itself back to the load value at zero (or -1?) Each time it hits zero, it sends a pulse to the lower 'LS191. Thus the lower one counts K times the upper one. However, it does not reset itself. Instead, the extra gates inhibit the output pulses once the lower counter hits zero, and prevent the lower counter from seeing more clocks until reset.

Cascading the counters will get you as many bits for N and K as you desire.

I can post the CircuitMaker file if someone wants it.