Take items off the kitchen counter and tape plastic sheeting over the counters and floor. Cut along the corner joint from the ceiling to the front face of the soffit using a utility knife. Your objective is to cut through the paper corner so that when you remove the drywall you will not damage the ceiling.
Cut along the vertical corners on each side of the face of the soffit. Cut along the bottom front edge corner. Inside corners are usually paper. Outside corners are usually metal. Bring the blade to inside edge of the metal to make your cuts. It is likely you will need to remove and replace the entire metal corners, but it is prudent to start by removing the drywall panel first and dealing with the repair at the end of the project.
Cut and remove the drywall using a drywall saw, a drywall rotary saw and a prying tool. Wear a face mask when dealing with drywall and plaster dust. Discard all of the old drywall. With the soffit open you will be able to see what is inside. If electrical fixtures are inside, you must provide shielding around the fixture boxes to keep the new insulation from touching and overheating the fixtures. In most cases, you will want to retrofit or replace the fixtures.
Cut batting-type insulation to fit inside the soffit, around and over any duct work and pipes. Some pipes might require pipe insulation if they are exposed to hard freezing conditions. Pipe insulation is sold in the pipe department of hardware stores and often is cut to size and slipped over the pipes. Wrap the pipes as required in your area for your conditions. Do not compress your insulation. Add insulation beyond the thickness suggested for your climate.
Close the soffit by removing any remaining drywall screws or nails and cleaning off the face of the studs. If you can attach drywall to a stud at the metal corner areas without disturbing the metal corner, you might be able to enclose the soffit quickly and easily. If not, remove the metal corners by prying them off with pliers.
Clean up the corner by removing any nails or screws and ensuring that the wood is smooth. Attach your new drywall by cutting the drywall to size. Screw the drywall to the studs using drywall screws slightly recessed into the drywall. Screw every 8 to 10 inches.
Measure and cut drywall paper inside corners and attach over the drywall using joint compound and a drywall knife. Apply compound to the drywall like a thin layer of glue. Press the paper into the compound. Add another layer of compound and smooth out the edges. Measure and cut the outside corners of metal drywall with tin snips. Attach the corners to the drywall with drywall screws every 6 inches. Apply joint compound over the metal and smooth it out with a putty knife. Apply compound over the screw heads and smooth out with a putty knife.