That is a beautiful garden.
A French drain would get rid of the water, but wouldn't necessarily put it in the beds, which is what it sounds like the OP wants.
I would dig tiny trenches across the paths at regular intervals in the hard pack under the
wood chips so that they run underneath the borders. The wood chips will hold pockets of space open for water to pass through the paths, into the drainage trenches (wood chip tiles?), and into the bottoms of the beds.
The woodchips will eventually decompose, so I would actually use terra cotta or unfired clay tubes, or pieces of broken clayware, to hold open the structure of the drainage trenches. If I had access to old-school terra cotta drainage tile, I would use those. Either way, I would probably augment the design for longevity with appropriately large-sized pebble fill to maintain the drainage trench structure, perhaps covered with a water permeable geotextile to keep the soil from filling the void spaces.
Or you could literally just design the borders so that every one in three or five boards are set slightly higher than the others, providing direct access to the soil in the bottom of the raised beds for water from where it pools on the hard packed soil under the wood chip path. This might be the easiest design to implement, but the drainage holes would need to be situated in dips of the hard pack.
There are options, some of them cheap and elegant in their simplicity, and others pricey but likewise elegant. Please keep us posted, or just point us to more of your beautiful garden shots, and good luck.
-CK
A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects.
-Robert A. Heinlein