First of all, try removing the battery and turning it on with only the powersupply plugged in.
If that doesn't work try to insert the powersupply and wiggle it a bit, if it feels like the part that the powersupply is plugged into (the dc connector) is loose or follows the power plug without resistance to the wiggle, it will probably need to be replaced or resoldered onto the motherboard.
If neither is an issue you might have to open up the laptop and take out any removable hardware and periodically try to power it on. if you get to a barebones motherboard with not poweron then it is probably the laptops motherboard that is defective.