How can I emulate the PS/2 protocol without a microcontroller?
You need to be careful: the PS/2 protocol is bidirectional, so you can't just transmit.  Once in a while the computer will pull the CLK line low and then drive the DATA signal for a few clocks, and then you need to acknowledge it.  Furthermore, your understanding of a keyboard protocol appears to be a bit incomplete.  They do not send ASCII values, they send key codes, including separate key-down and key-up events.  Also, you need open-drain outputs.

It would be far easier to do this with a microcontroller.  Regarding your specific arguments against them in your comments:


Cost: there are lots of microcontrollers with enough I/O to read the key matrix and speak PS/2 that are less than $0.50, like the PIC16F54.  You will be hard pressed to get all of your discrete parts for less.
Programming: You can order chips pre-programmed with whatever firmware you provide from a distributor like Digikey, or directly from the manufacturer.
Efficiency (space): A single microcontroller will be much smaller than several discrete chips and usually includes an internal clock, pull-ups, open-drain buffer, etc. (Note that open-drain can be achieved by switching a pin between two states: output 0, or input with pullup enabled.)
Efficiency (power): Microcontrollers have lots of sleep modes available and can be almost completely powered off while no keys are being pressed, or in between scans of the keyboard.
Efficiency (your time): You will likely run into lots of problems besides the protocol issues, like needing to debounce keys, etc.  Firmware fixes are going to be a lot faster than having to respin a PCB or order some new chips.