Backyard Drainage Issues
I had nearly the same problem and a single french drain made the problem go away. It is important (and probably worth reinstalling if not done) to have:


a 10% grade (1 inch down for every 10 inches of pipe length)
cloth covered french drain pipe
plug off the yard-end or open it to a ground level drain (cloth and grate covered!) and make sure you can access it to clean it out
make sure your trench is at least 8 inches wide (2 inches clearance on either side of a standard 4 inch french drain pipe)
place the french drain pipe in a channel that has 1-2 inches of pea gravel for a bed
cover the pipe with at least 2 inches of pea gravel before putting in the dirt


The french drain also should start prior to the onset of the wet area.
(Please note the image does not accurately reflect the opening of the yard end - i.e. you can't clean this one out!)