Trying to restart a MacBook Pro that went off by itself with a battery that indicated "Service Battery"?

My daughter's MacBook Pro (Late 2011 model) just went off while she was using it.

Battery was reporting that it required servicing, so apparently, she was pushing it.

After it went black, it would not turn on anymore, even though it said the battery was full (before it went off) and the charger was plugged in.

Now when the charger is plugged in, the charge light flickers very faintly or is a constant faint green (really faint). The charger is good because it can charge my son's MacBook Pro just fine and shows a bright orange while charging.

We've tried long power button holds to force a restart, but nothing seems to work.

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Update: MacBook Pro was brought to an Apple store. They swapped out the battery and put in a new Magsafe connector which they assumed might be faulty. Neither of those fixed the problem. So they suspected it was the Logic Board and said it needed to be sent out.

This MacBook Pro was sent out previously and received a logic board replacement earlier in 2014. Now it's potentially going to get another logic board replacement. We told the representative at the Mac service desk (aka Genius Bar?), and they said something about there not being a warranty on the logic board and they don't have a "defined lifetime". If it is a logic board, that will make it the third logic board that's been in this MacBook Pro.