The parts that are broken.
How do I know? You're in front of it!
The way this works, with i5/i7 machines, they turn themselves on after the charger is plugged in. The old $@$* doesn't. So what you'll have to do is find a keyboard/trackpad/trackpad-to-mobo-cable that works setup so you can see if the board itself is capable of turning on.
This is NOT before cleaning it, in an ultrasonic cleaner(SWEEPING FREQUENCY NOT JEWELRY CLEANER) with a gallon of distilled water, 103 milliliters of branson EC, and tossing it into 99% alcohol before letting it dry for a day.
Then measure power rails when plugged in. I am assuming your DC In flex is not corroded. PPBUS_G3H, PP3V42_G3H, PP5V_S5, PP5V_S3, etc, and report back with some findings.