In a medium stockpot, combine 4 cups of water and the kombu and bring to a boil. Using a slotted spoon, remove the kombu. Add the bonito flakes and remove the pot from the heat. Allow the flakes to settle to the bottom of the pot, and strain the broth. Add the soy sauce, mirin, and lime juice.
Divide the noodles among 4 bowls. Divide the vegetables evenly among the 4 bowls. Pour in the dashi, top each bowl with a little grated ginger, and serve immediately.