I don't have any experience with the A1278 aluminum unibody MacBook, so I don't know what's going on for certain; it could be an issue with the MagSafe board, an issue with the motherboard itself, or even the power button. And no, A1181 parts won't work in the A1278.
One of the best ways to diagnose power-on problems is to hotwire your Mac by getting at the power pads, two small contacts on the motherboard that you can jump to instantly power on your MacBook:
Beetstech Repair shows you how to use the logic board power-on pads to boot a Unibody MacBook Pro without the power button. Plus photos of where to find power-on pads on the logic boards of the Apple MacBook Pro 13″ A1278 (2009, 2010, 2011, 2012), MacBook Pro 15″ A1286, & MacBook Pro 17″ A1297.
beetstech.com
Though I suspect that the problem may be an issue with the motherboard's power circuitry or the MagSafe board. Perhaps your Mac was previously a victim of liquid damage.