Break heads off sardines and remove the small black pellet of innards just beneath each head (they're dried, so this isn't messy). Put cleaned sardines, mushrooms, kombu, and 8 cups water into a medium pot. Cover and chill overnight.
Set pot over medium-low heat and bring to a simmer. When small bubbles form along sides and bottom of pot, but before it actually begins to simmer, remove sardines, mushrooms, and kombu.
Increase heat to high and bring to a boil. Add bonito flakes, pressing down with a spoon to submerge, and return to a boil; then immediately turn off heat. Let flakes settle to bottom of pot, about 15 minutes.
Strain broth through a cheesecloth-lined strainer into a bowl. Wring out all liquid from cheesecloth into bowl. Rinse pot, pour in strained broth, and heat over medium-high heat. Add mirin and soy sauce; simmer 5 minutes.
Make ahead: Up to 2 days, chilled.