Make the brine: Bring water to a boil in a large pot. Add salts, sugar, and spices; remove from heat, and stir until salts and sugar dissolve. Let cool.
Make the corned beef: Place brisket in a nonreactive container just large enough to hold it. Pour cooled brine over meat. Place 2 small plates on top to keep meat submerged; cover, and refrigerate for 2 weeks.
Rinse brisket; discard brine. Place in a large pot. Add enough water to cover by 2 inches. Add onion, celery, and halved carrot, and bring to a boil. Reduce heat, cover, and simmer until very tender, 3 to 3 1/2 hours.
Set a steamer in a large saucepan. Add enough water to reach the bottom, and bring to a boil. Add turnips. Reduce heat, cover, and steam until tender, 8 to 10 minutes. Transfer to a bowl. Repeat with baby carrots, steaming 10 to 12 minutes. Add to turnips.
Transfer corned beef to a cutting board. Tent with foil, and let rest for 30 minutes. Discard remaining solids from broth, then bring to a boil. Add cabbage and potatoes, and simmer until tender, about 25 minutes. Add turnips and carrots, and cook until warmed through. Transfer vegetables to a platter; reserve broth.
Trim excess fat from beef. Slice thinly against grain, and transfer to platter. Serve with broth and mustard.