Hi!
Does the power adapter have any sort of status LED? Like one that’s supposed to light up when you plug it into the wall?
Do you have a multimeter to measure the current supplied by the power adapter? <— this could quickly tell us where the problem is. If not, I’d try to get a new power adapter first, as laptops should boot even with a bad/ non-existing battery - but if the battery is empty and the adapter broken it’s clear what it will never work.