Heat the chicken stock in a medium saucepot over medium-high heat. Peel the daikon, trim the ends and cut into halves or thirds to create round chunks that will fit in the saucepot. Place the daikon in the stock and season with 1/2 teaspoon salt. Lower the heat and cook the daikon at a very slow simmer until it is tender, but has a slight resistance when poked with a metal skewer or the point of a knife, 20 to 30 minutes.
While the daikon is cooking, slice the radishes and the green part of the scallion very thinly and immediately place in a bowl of cold water. Remove the daikon from the chicken stock and set aside. Add the cilantro to the stock and simmer for 1 minute, just to flavor the stock. Strain the stock into another pot, discard the cilantro and add the soy sauce. Keep the stock warm.
Strain the kimchi over a bowl to reserve the juice. Chop the kimchi into bite-size pieces and set aside.
Slice the daikon into 1/3-inch-thick rounds and arrange them in a ring around the bottom of 4 shallow rimmed bowls. Place about 1/4 cup of the chopped kimchi in the middle of each ring made from the sliced daikon. Garnish the top of the kimchi with some slices of radish and scallions. Gently pour the hot broth into the bowls just to barely cover the daikon. Sprinkle a little of the reserved kimchi juice over the daikon and serve immediately.