SteveG47 over at Microsoft Answers provided a solution.
- Go to
Control Panel\Hardware and Sound\Power Options\System Settings
- Click on Change settings that are currently unavailable
- Uncheck Turn on fast startup (recommended)
- Restart your computer
Below is how the Recording tab in the Sound control panel looks when the Surface Pro 4 Microphone is working properly.
Note the level indicator on the right. When the mic is not working, that indicator is either absent or unresponsive.