It sounds like you have two issues here:
You state the system does run working from the power adapter. So do you see the faint image of the desktop on the display? Use a flashlight to shine tough the lids Apple logo (a poor mans backlight), are you able to see the desktop icons? If you are then you know there is an issue is within the backlight circuit. Which could be a fuse, bad cable, WLED or inverter depending on the model.
Your second issue where your system won't work with the battery alone tells me you have a power control issue which often comes down to a burn't FET or comparator in the charging circuit.
But without knowing the exact model we really can't answer your question.
The 820-2936-b logic board is not from a MacBook but a MacBook Pro.