Wrap the niboshi in cheesecloth and tie the edges together with kitchen twine to make a little satchel. Add the satchel along with the ginger, garlic, water, mirin, sugar, sake, soy sauce, and 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 pork belly. Turn down the heat to low, cover, then simmer for 2 hours.
Add the turnip and cook until the turnip is 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.