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.