Heat the oven to 325°. In a small saucepan, bring the kombu and 1 1⁄2 cups water to a boil. Remove from the heat, remove and discard the kombu, stir in the bonito flakes, and let stand for 5 minutes. Pour the dashi through a fine sieve into a bowl and let cool.
In another bowl, whisk the mirin and soy sauce with the eggs, then whisk in the cooled dashi. Divide the custard among four 6-oz. ramekins set inside a 9-by-13-inch baking dish, and set the baking dish in the oven. Pour enough boiling water into the baking dish to come halfway up the sides of the ramekins and bake the custards until set but still slightly loose in the center, about 30 minutes.
Remove the baking dish from the oven, lift the ramekins from the water bath, and transfer to a rack. Let the custards cool slightly before serving.