That blocky device that plugs into the wall is the POWER ADAPTER. I am not trying to be a jerk about terminology, what these boxes do and how they do it is essential for understanding how they all work together. The LEDs in the MagSafe end are controlled by commands from the computer.
The CHARGER is inside the MacBook Pro, and is completely under program control. Your Mac accepts some power when needs to run, and accepts more power when intends to charge its internal battery. A power source like the Power Adapter can not 'force itself' on your MacBook Pro. Only the algorithms inside your Mac can decide when is a good time to charge the battery.
When the battery charge level goes to zero, strange things happen. "smart" devices may not wake up properly and cooperate. I suggest you connect the power adapter and shutdown your Mac, and let it try to trickle-charge all night long. In the morning, try resetting the SMC:
Reset the SMC of your Mac - Apple Support
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