I have the exact same problem. See screenshot below. My surface pro 3 dies instantly the charger is removed. See the grey, hollow battery icon in the tray in the screenshot which is empty which implies that the battery is not connected. I can only use my surface pro plugged in which is hopeless.
Perhaps Microsoft could confirm the death circle whereby windows (firmware?) updates to fix the battery problem will only install when the battery is charged to 25% but the battery cannot charge to 25% because a previous windows update has screwed up the firmware supporting the battery so the firmware will not install.
Any fix for this battery problem would be greatly appreciated.