To answer everyone’s question about why Flexing works: My best guess would be that the flexing forces back a poorly-designed connector to re-establish a proper connection. The connectors inside the surface or tiny, and even the tiniest of movements can make them out of alignment, or in this case, return to its correct alignment. Due to the nature of its otherwise complete lack of response, I feel that the connection that is being made is the one involved in transferring power from the battery to the motherboard of the Surface Pro 3.

Lol. Who knew, the frustration factor would reign victorious! Thank you for this gem of a solution. Worked for me, as well.