Hi, and welcome to the forums!

This is just my experience and I'm not ruling out other possible causes. Sometimes a fuse or inductor on the +5V line to the keyboard connector can fail to a high-resistance state. Different keyboards have different current draws, and can be more or less tolerant to out of spec supply voltage. Had that happen on a Abit KT7 that was donated to me, previous owner said the keyboard doesn't work, but for me it did. Puzzled, I made some measurements, and turned out that only around 3 volts reached the keyboard and the difference was lost on the failed inductor. So my keyboard was happy with 3 volts, but I guess most others would stop working at such a low voltage.

Thanks so much for the answer, I think you figured out the cause.
I measured the output from the AT port (+5v and GND pin of course) and it was zero... so perhaps the chinese keyboard was able to work even without 5 volts and indeed the leds were always off when it was connected to that computer.
So I connected the CLK and DATA wires to the AT pin and +5V and -5v to the molex red and black cable. To do so, i modified and soldered wires in the PS/2-AT adapter. All connections were fine but unfortunaley I did a terrible mistake: to look better at the monitor I moved the keyboard and two wires short circuited, my guess is that CLK and DATA shorted because the power wires were insulated very well. Smoke and an horrible smell came out from the keyboard and of course it was dead even on another computer. I am a bit sad because it was the only keyboard I have used in the past 15 years...
As a matter of principle, I really wanted to solve the issue today so I figured out another solution: this time I used a PS/2 => USB connector, a USB dongle and an AT cable from a broken AT keyboard (it was shortcircuited probably due to leaking capacitors). In this way, I connected CLK and DATA to the mobo and the +5 and -5 to the usb connector plugged inside a phone charger.
This time I checked the connections several times with the multimeters and insulated everything with a lot of tape. The only keyboard left was the Model M and I really not wanted to fry it.

The keyboard still is not recognized but an important detail has changed. Before doing the external power thing, if I connected the Model M to the mobo it remained with led off and nothing happened if I pressed the key. Now leds are on (due to usb charger) and, the important thing, every time I press a key the motherboard emits a beep. So it is receiving some input from the keyboard but it is not recognizing it as a keyboard. Can I move forward at this point in some way or do I have to accept that today I've lost my keyboard, my Pentium I and all my retrocomputing projects?

EDIT:
It's working!!!
I have found this post on vogons: Keyboard error / no keyboard present
where a user reminds that there is often a continuity check between the outer ring of the connector and the keyboard. I have connected the outer ring (At => Usb insulation shield) but the crappy PS/2 to USB adapter no, so I put a small wire between the ps/2 connector and the USB connector and now it works perfectly and I am very happy, I hope I won't broke everything when moving the "thing" from a temporary to definitive state. Moreover, I want to get rid of the phone charger. Is it safe to connect the 5V directly to the Molex? I am really scared that the thing happened to the old keyboard could happen to the Model M too.

Good work!

As for that PS/2 model M, no surprise that was problematic. IBM keyboards were great, but with great build quality came great power draw, a Model M draws about ten times as much as a contemporary Alps-based keyboard. So if any board is going not to work in a case like this, it's one of those monsters.

manalog wrote on 2021-01-24, 20:21:

Moreover, I want to get rid of the phone charger. Is it safe to connect the 5V directly to the Molex? I am really scared that the thing happened to the old keyboard could happen to the Model M too.

Yes ! The +5v on a Molex is the same as the +5v on the AT PSU cables where the KB power comes from on a proper working one.
Example: On one my Pentium board get a reading of ~0.4 ohm from the AT PSU +5v connections to the Keyboard +5v pin #5 on the AT KB connector, which is good cause only the little fuse is between the PSU line and the KB.
Too bad about the old KB getting shorted. It is getting harder to find any decent PS/2 KB's in my area, No true AT found in nearly a year. Last ones found were two HP black/silver (kb-0316) ps2 for $3 each and they work well for test KB's.