Peel and devein shrimp, leaving tails intact. Cut fish into bite-size pieces. Remove stems from mushrooms; halve or quarter large caps.
Tie kale stems together with kitchen string; blanch in lightly salted boiling water for 10 seconds. Drain; rinse under cold running water. Squeeze water from kale. Trim off tied stems.
Cut kale bundle and scallions into 2-inch lengths. Drain noodles and cut into 4-inch lengths.
Mix dashi, mirin, and soy sauce in flameproof casserole. Heat to boiling, then reduce heat to maintain steady simmer. Add scallions, carrots, and kale; simmer 3-4 minutes. Add mushrooms and then pieces of fish, one at a time. Simmer 1 minute, skimming any foam from surface. Add shrimp; simmer 30 seconds. Remove from heat. Stir in noodles. Cover pot and bring to table.