Hi all,
I wanted to weigh in here as I had the same issue. None of the steps I found here or anywhere else on the internet worked. What did work for me was to use the non-dock type surface pro charger that plugs directly into the wall socket. I plugged it in and saw a solid white light on the connector straight away, turned the Surface Pro on 10 minutes later without issue. I suspect the charging dock could not provide sufficient charge for the Surface Pro with a completely dead battery. Anyway, hope that helps someone out there.