One would
think that an LED on the PS would be to indicate it was working properly. OTOH, it
is a PC.
However, if you can disconnect the cable going from the PS to the motherboard, you should be able to use a simple volt meter to check the volatge and polarity on its pins. That might at least show the PS is OK.
As for the excess Goggle results, you might try adding "HP" "no power" "power light" etc. to the search criteria and get fewer but more meaningful results.
Of course, there must be a dozen or so computer shops around your area that can quickly and with a small fee determine that you need to buy a new box from them!