Wrap the 8 small niboshi in cheesecloth and tie the edges together with kitchen twine to make a little satchel. Add the satchel along with the 5 rounds ginger, 2 cloves garlic, 1 cup water, ¼ cup mirin, 1 tablespoon evaporated cane sugar, 2 tablespoons sake, 1 tablespoon soy sauce, and 1 teaspoon kosher salt to a small dutch oven or other heavy pot with a lid. Heat the pot over high heat until it comes to a boil, then turn down to medium low and simmer for 15 minutes.
Remove the satchel of fish, then add the 1 pound pork belly. Turn down the heat to low, cover, then simmer for 2 hours.
Add the 3 medium turnips and cook until the turnips are tender and the pork belly falls apart when prodded with a fork (about another 30 minutes). Gently transfer the pork belly and turnips to a bowl, then skim all the fat off the braising liquid. Strain the liquid over the pork and turnips and serve with hot rice.