Desktop

Of any LED's on the M'Board lighting up but as you have only provided such sketch information and no information on what if any fans are running I would say that the most likely thing is that the M'Board has been damaged and needs replacing.

This most often happens with Cheap Power Supplies that offer no protection to the Internal Circuitry of the computer and cause the internal Circuitry to be destroyed in the few nanoseconds between the power coming back on and having a solid stead supply.

What most often happens with Mains Outages is that the mains drop out and then 15 seconds latter you get a short return of power and then a long outage at the first restart of power most of the damage is done.

Col

Depending on the mobo there is usually 5 or more capacitors located right beside your cpu. If any or all are puffed then thats another good indication your mobo is dead. In which case there may be a good chance at needing a new cpu as well

It also wouldn't be the first time that someone has bought a faulty piece of equipment from a store. See if someone around you has a spare that is running for sure (make sure its compatible with your mobo of course - wouldn't want someone who has an old at system kicking around still and trying to use that one) and test with that one.

I know that by buying a new psu you've bypassed this issue but something I've noticed in my experience limited as it is some motherboards do have some form of internal reset where after a power outage it won't reboot but if you disconnect the psu from the mobo and plug it back in, if the rest of the hardware is good, it will boot up.