Heat the 3 ¼ cups water with the kelp in a deep saucepan. Just before it boils, remove the kelp and pour in the dried bonito flakes. Boil, stirring, for about three minutes, then strain into a clean deep saucepan. Add the soy sauce, sugar, sake, mirin, and salt.
Bring this sauce to a low boil and add the drained lotus root and gobo, and the carrots. Cover and cook over medium-low heat for about 12 minutes or until the vegetables can be easily pierced with a toothpick. Take out and set aside to cool. Add the shiitake mushrooms to the sauce, cook for 12 minutes, then remove. Next cook the konnyaku for 10 minutes; take out. If the sauce has reduced too much, add a little more water. Simmer the snow peas for just 2 minutes, and remove. Finally, cook the drained sato-imo for about 12 minutes (or to your preference), turning once; test softness with a toothpick.
To serve, place the vegetables in separate decorative mounds on a serving plate or bowl. Garnish with snow peas and serve at room temperature as a side dish.