Combine soy sauce, 1/2 cup water, brown sugar, honey, mirin, ginger, garlic, and red pepper flakes in a large bowl to make the sauce.

Place drumsticks into a multi-functional pressure cooker (such as Instant Pot®) and pour sauce on top. Close and lock the lid. Select high pressure according to manufacturer's instructions; set timer for 15 minutes. Allow 10 to 15 minutes for pressure to build.

Release pressure using the natural-release method according to manufacturer's instructions, 10 to 20 minutes. Remove drumsticks and set aside.

Switch to Saute function and bring sauce to a boil. Mix 2 tablespoons cold water and potato starch in a small bowl to make a thick slurry. Quickly stir mixture into the sauce and continue stirring until sauce has thickened to your preferred consistency. Serve drumsticks with sauce.