It seems you’ve tried everything except resetting the SMC (System Management Controller). Here’s a link to Apple’s article on this topic. https://support.apple.com/en-us/HT201295 Good luck!
I realize that this is a very late response, but perhaps it will help others who search the "Answers Index" area here. Have you tried a(n) SMC reset? This can "cure" a lot of issues. (1) Reset SMC of an iMac, Mac Pro, Mac Mini: (2) Shut down the computer. (3) Unplug the computer's power cord. (4) Wait fifteen - twenty seconds. (5) Attach the computer's power cord. (6) Wait five seconds, then press the power button to turn on the computer. OR (1) Shut down your Mac (2) Disconnect the power cord (3) Wait fifteen - twenty seconds. (4) Press and hold the Mac’s power button for 5 seconds (5) Release the button (6) Reattach the power cables and boot the Mac as usual
N.B.! MacBook Pro Early 2011 (and presumably all 2011s): the Battery Connector has no pull tab; it’s a pin-type connector that you must detach by pulling UP (perpendicular to the logic board i.e. towards the ceiling). I had to use a pry tool - I gently pried the left & right (lateral?) ends of the connector upward & away from the logic board to loosen it. I then pulled it off the board gently using my fingers.
