Do I need to replace my battery?

Hi all - I recently was given a late 2009 Macbook with the polycarbonate body. Was told it no longer turns on. Opened it up and didn’t see any noticeable damage.

I finally got the correct charger for it, but when I plugged it in, the green light on the charger does not turn on. This is brand new charger, so it’s unlikely that the charger is bad (but possible).

Would this mean a faulty battery? Any troubleshooting I can do to get this thing to charge?

Thanks!