Might be a blown fuse on the motherboard. Verify that you are getting power from the power pin on the PS/2 port.

What mouse are you using?
I just bought an Asus P5A rev 1.04 last week, and I had the most bizarre issue: Windows would freeze during boot (at the loading screen), or, if I tried to install it from scratch, Windows setup would also freeze even before starting (at the "Copying files needed for Windows" step)

I basically lost a whole day trying to debug the issue, replaced (almost) EVERYTHING multiple times (video cards, RAM, HDDs of various capacities & PATA cables), but the problem would not go away.
Finally, I started thinking outside the box, and wondered if I was using any device which was not "period correct" for this platform... and immediately I thought of the PS/2 optical mouse.
Removed the mouse and I finally reached the Windows 98 desktop (and even Windows setup was loading properly).

After that, I replaced the optical mouse with a 20 year old classical mouse with a ball, and everything works great.
What's strange is that I used this optical mouse with HUNDREDS of motherboards (socket 3, socket 7, super socket 7, etc) and I never saw anything like this.
My guess is that this board is not able to deliver the power needed by a more "modern" mouse, which is why it can behave erratically.

Anyway, not sure if your situation is the same or not, but thought I'd just mention the problem I had, maybe someone else will face a similar issue 😀

If your PS/2 port is really broken, you can get an adapter for your serial port to make a PS/2 mouse work there instead:
https://www.serdashop.com/PS2TOSERIAL

Thank you for your input, guys!
I finally found some time to have a closer look at the board again and discovered that there are some damaged SMD parts that I didn't notice before.
Measuring the voltages, it turned out that the PS/2 port voltage varies between +1.5V and +3.5V depending on the combination of different keyboards and mice that I plug in (the optical mouse that probably draws most power causes the voltage to drop to +1.5V). The USB ports get nice and stable +5.0V and both fuses are OK.

In the above picture, R11 is defective. (R12 is missing, but looking on some hi-res pictures of the board online, it seems that it wasn't populated in the first place.)
Capacitors C27 + C28 and coils L9 + L10 are damaged, too. These clearly belong to the USB ports and shouldn't have any impact on the functionality of the PS/2 ports, but it explains why Windows wouldn't detect a USB mouse either.

The parts near the PS/2 ports look OK though.

So I'd say a likely cause for the voltage at the PS/2 ports being too low is R11.
I'd like to go ahead and try to replace all the parts, but the challenge is that all the values are unknown... I am not an electrical engineer or something, so I probably wouldn't succeed in reverse-engineering the circuits to determine the correct values. All I can do is probably try out some random values and see if it works. *sigh*

The ones designated with L are not resistors but inductors.

I can measure the resistors on one of my boards.

On my board R12 is marked 564, which is 560kΩ. For the inductors I'd use 1µH as their value is not critical.

I replaced the caps and inductors near the USB port with some that I salvaged from the USB circuitry of a dead Mobo so the values should be more or less similar.
R12 was mechanically damaged and when I desoldered it, the solder pad came off so I had no chance to solder in a replacement. Since R11 seems to be in parallel to R12 though, I populated it with the closest match that I had at hand, 120 kOhm.

Nothing had changed in regards of PS/2 voltage being too low, so I poked around with the multimeter some more and discovered that L1 caused the voltage to drop from 5V to something between 1.5 and 3.5V depending on the devices connected. So I desoldered it and replaced it with an inductor from the PS/2 power supply circuit of the donor board - voilà, perfectly stable 5V at the PS/2 ports. Upon desoldering, L1 had broken into two parts immediately, so it must have been cracked before, causing the voltage drop.

Now the optical mouse lights up just fine, but still isn‘t recognized by CuteMouse or Windows. Meh. Might try an older ball mouse, but usually the optical one works with PCs of this vintage just fine.
Edit: 1999 Logitech ball mouse doesn't work either.