Dug finally rerouted the kitchen drain. To give you an idea of what we were dealing with, I've provided illustrations.
The first is a diagram of how water drained from the kitchen sink to the sewer line. Notice the many-cornered path the waste water had to travel. This route + corroded pipes = eventual slowing of drain.
The second diagram illustrates the new route. Straight lines, few corners. Easy peasy.
Not only does the kitchen sink have happiness, but our downstairs (and right now, only) shower can drain by itself. The kitchen sink drain pipe used to run into the shower drain and often plugged it. Few things are grosser than taking a shower that stops draining and suddenly pieces of decomposing food start floating around your bare ankles. Nastiness.
But now it's all better. One more step to a completely happy house.
For your viewing enjoyment. Here is one last picture. This is a cross-section of a piece of pipe that Dug removed. Believe it or not, this is not the worst of them. (Sorry I didn't add anything in the picture for reference. That hole is just big enough to let a nickel fall through.) And we wondered why the sink and shower were always clogging...