Kombu (a type of Japanese seaweed) and bonito flakes (dried-fish flakes) are available at Asian markets.
Bring 6 cups water, mushroom stems, dried mushrooms, and kombu to a boil in a large pot over medium-high heat. Remove from heat, and add bonito flakes. Let stand until flakes sink to bottom of pot, about 3 minutes.
Line a sieve with cheesecloth. Strain broth through sieve into pot, and discard solids. Stir in soy sauce and mirin. (You should have 5 cups.)
Cook sweet potato in a large pot of boiling water until crisp-tender, about 2 minutes. Transfer to a bowl using a slotted spoon. Add noodles to boiling water. Cook according to package directions. Drain. Rinse with cold water.
Divide noodles among 4 bowls. Arrange sweet potato, sliced mushroom caps, snow peas, kohlrabi, and tofu on top of noodles. Bring broth to a simmer, then immediately divide among bowls. Serve immediately.