Nick:
We have a similar climate here on the west coast of Canada. You can calculate the BTUs required to heat your lean-to at this site:
http://www.littlegreenhouse.com/heat-calc.shtml
If it's the shape I assume it is based on the dimensions you've provided, the area inside your green house would be somewhere between 50 and 60 cubic ft.
With your minimum temperature of 16F, assuming that your greenhouse is covered in 4 mil polyethylene, and that you want a temperature in the greenhouse to remain at 32F when it's 16F outside, you would need a heating device that puts out about 1,000 BTUs according to the web site above.
I just strung up a 15 foot string of some old style (not LED) Christmas lights in my mini greenhouse (a string of 20 lights). At 2 ft by 2 ft by 28 inches high the capacity of my mini greenhouse is only 21 cubic feet, or about 1/3 the capacity of yours. The outside temperature right now is 33F, and I turned the lights on to see what the temperature would be inside the greenhouse. After waiting a couple of hours it now reads 60F, so I can calculate the heat produced by the Christmas lights, using the formula on the web site above, at about 680 BTUs.
Making the conversions for your sized lean-to, my set-up would give you a temperature inside of just over 27F when the outside temperature goes down to 16F, and would keep your plants above freezing if the outside temperature stays above 22F. You could increase the heating capabilities by either adding more Christmas lights or using a cover that gives a lower heat loss value.
The lights cost about $10 Cdn (₤5)
Here are some pictures of my set up: