try making sure your D-Pad ribbon cable is secured in because that allows for power transfer to the power switch (psp 3000 and 2000 model ribbon cables have diffrent power flows so make sure it's a psp 3000 cable). also make sure that nothing metal is stuck in there and is touching the motherboard.
also if it was a brick(which it is clearly not) the light would turn on but nothing would happen (i like to call it the "forever light")
if all else fails, completly dissasemble the psp and then reassemble it. some parts can be loose or not connected (EX: forgot) and reassembling the psp might fix the problem.
Oh and before i forget... make sure all the screws on the casing are tight (especially around the battery compartment) and try changing the battery.
hope this helps!