Repair or not
First consider whether any repairs should be carried out at all. Repairing can sometimes cause more damage, and if it is an older piece, finding an exact match may be impossible. So gluing may be the better approach.
If small shards of glass are missing, daylight is visible through the broken part, or sharp bits are exposed, then it should be repaired, assuming the client is willing to pay.
Starting the repair
Having decided to repair, the first action is to use fine steel wool to remove any patina from the solder around the broken pieces, on both sides.
Removing the broken glass
Next, work out the shards using a fine blade to get between the broken pieces. You should be wearing safety glasses during all the processes of getting the glass out of the piece.
In some cases in may be necessary to score the remaining part of the broken piece with your cutter in a criss-cross pattern, from edge to edge. Also try to go toward the centre of the piece from each corner while making the score lines.
Now start tapping the centre of the broken, scored piece of glass with the ball on the end of your cutter and then work out to the edges. Keep tapping the glass and it should start to fall out. When most of the glass in the middle has fallen out, take out small pliers and gently wiggle the broken shards out, one at a time. Remove all the glass possible, but you might not be able to get the glass in a tight corner.
Removing the solder and foil
Now take your soldering iron and melt off the bulk of the solder seam, slowly going all the way around the hole. Wipe excess solder off on your iron stand sponge. Do the same on the other side. During this process you will be able to remove any remaining small pieces of glass.
Apply the hot iron to the corner joint to expose the piece of copper foil that was wrapped around the broken piece of glass. With a pair of tweezers, grasp and very gently tug out that inner piece of foil while keeping the iron on the solder line to keep the solder molten. Try not to pull the foil off adjoining pieces. If some other foil does come off, then remove that entire length, scrape clean, wash off, dry, and re-apply a piece of the same type of copper foil, cut to size.
Re-flux the exposed foil lines, remove any solder blobs - particularly in any corners - with your iron, wiping excess off on your sponge again.
Replacing the glass
Taking the shape
Take your piece of glass that best matches the broken piece, and position it underneath the hole. Trace the shape of the hole onto the glass. This works best for flat panels. Curved forms usually need to have a paper template made by tracing from the back onto the paper. Cut the shape out of the paper and put it on the piece of new glass. Make sure that the grain of the new glass is in line with the original. Cut the piece out, grind to shape, doing lots of test-fitting in the hole, until it fits.
Clean any flux off the new piece from your test fittings, and apply the same size and colour backing of copper foil that was used on the piece that you removed.
Fit your replacement piece into its place, making sure that it is level with its surrounding pieces. You might want to tape it into place. Check its position, and if you are happy with it, then tack it twice per side.
Proceed to solder it into place. If your fit wasn't perfect, or the piece is a little rough, fill in any gaps with some 50/50 solder. Let it cool, and then finish off with 60/40 solder. Blend your new solder lines with the quality of the solder lines on the rest of the piece.
Wash off the flux, scrub clean, and patina if required. If copper patina was used on the piece, don't worry when you apply the copper patina. It will not match if the panel is not brand new as copper patina ages much more than black patina. Wash the patina and polish it to blend the piece in.