Chosen Solution
MacBook Pro a1278 i5 stopped turning on I changed the battery and it’s working again, but it’s not charging the battery, what could it be?
It may be a problem with the motherboard or screen. Have you checked out the troubleshooting page from Ifixit? MacBook Pro 13" Unibody Troubleshooting
I would check the battery using this gem of an app CoconutBattery post a snapshot of the apps main window. Also check the onboard battery checker on the side how many LEDs do you see and are they blinking?