Check out monoprice ps2 adapter on amazon, they are cheap and work great.
https://www.amazon.com/Monoprice-Keyboard-Converter-Adapter-110934/dp/B00IACID2C
This is the ps2 converter I have: https://www.amazon.com/dp/B00IACID2C?ref=ppx_pop_mob_ap_share
I apologize if my suggestions sound dumb:
if you have access to a different computer – assuming your device is a computer – perhaps you could try a laptop or one with a different OS like a macbook. That would tell you if it's a Windows issue or not.
If you can use that same adapter and plug in something else with a PS/2, you could try to rule out if it's the keyboard that's the problem. Maybe you have a mouse lying around or you could go to a local hardware store if they have some old tech you could try. The lighting up might be a good sign but it could be something else on the PCB too. Is the keyboard new or used?
If there's a way you can buy an another PS/2 to USB converter from say Amazon, perhaps you could try it and see if it works. If it doesn't, Amazon often has free returns or you're only out another $10ish.
FWIW, I have this one and from an old reddit thread you might've already seen, someone recommended this one.
AT > PS/2 then PS/2 > USB
e: change USB converter