Need advice on a strange light instalation.

So my Grandmother wants me to put a new light fixture in her kitchen, here's the situation.

There is an old light light fixture above her sink. It is pull chain operated so obviously always hot. In front of the sink is a window. Slightly below and in front of the light is a weird piece of wood I can only describe as a "floating board". It runs between her cabinets and is only anchored down at the two points where it meets the cabinets. It sits about 6 to 8 inches away from the wall, obscuring the light from the old fixture, hence the need for a new light.

She wants me to install the new fixture on the actual piece of wood if possible. However, I'm not sure if code would allow that since I won't be able to put the wiring in a box, not including the fixture's housing of course, I would have to drill a hole and run the wiring through it.

The new fixture does not have a pull chain by the way, so I will either have to jump off the old fixture or run a switch leg, however, I believe the window will make it too difficult to run a switch leg because I'm sure there is a stud or two in there somewhere. There is a crawlspace in the ceiling, but no attic, so I don't think I will be able to fit in there to run wires above the studs.

So uhh, any ideas? I may be able to provide pictures in a couple of days, I need to get back out there soon to look at a few switches she says aren't working right.