To turn it off, reduce the brightness of the back light to 0 by hitting the F1 key repeatedly.
To turn it on, and increase illumination of back light hit the F2 key repeatedly.
If your Surface Pro 3 keyboard will not work (touchpad and keys) then of course you try the obvious:
1. Make sure your computer has the latest Windows updates.
2. Gently clean off the connectors with a cloth and a small amount of rubbing alcohol.
But this evening, neither of these options worked. After rebooting several times, I went back to Google to search more possible fixes and discovered this one.
3. Make sure the Surface is turned off. Press and hold Volume Up (on the left side) and the Power button at the same time for at least 15 seconds. You may see the screen flash the Surface logo but continue to hold for 15 seconds. Release both buttons and wait 15 more seconds. Turn on the Surface as usual.
The third option above worked just fine for me this evening.