This can be fixed but it is
extremely difficult. If the pin is broken down below 'flush' on the back of the diode, the deeper it's down in there the better. You need a
1/64" drill bit which is about as thin as the diode pin. Using as much caution as possible and
very little pressure, try to drill out the top of the broken pin as much as possible without drilling into the center of the diode. You're trying to grind the pin down so there's a small hole where the pin once was - without completely drilling out the pin. This process took me about 45 minutes before, that's how slow & careful you have to be.
Once you've made that small hole that's only a few millimeters deep, get yourself a junk electronics circuit board & a pair of needlenose snippers. Looking at the components on the circuitboard, try to find something that uses pins that look similar to the diode pins in thickness. This is trial & error, it's hard to be exact - but one of the legs on a capacitor, transistor or microchip should eventually size out pretty close. What you're looking for is a replacement pin off of something, and you're going to need it to be at least an inch long. Once you find a suitable replacement 'pin' that's long enough (doesn't matter if its too long, you can cut it later) - you need to use the tiniest amount of 5min epoxy inside the hole you drilled & mount/balance your replacement pin.
DO NOT use any other adhesive but epoxy! Epoxy dries with 'elasticity' that allows a little movement/flexibility after bonding, anything else & the pin will just slide right out of that microscopic hole you just painstakingly drilled.
Once you've gotten the pin balanced & bonded, you can now solder the base of the pin to the diode case. If you can manage to do this without soldering anything else on back of the diode you will have fixed your problem. Now before you try to solder a new lead on the replacement pin, you need to secure it so you can solder
onto it without unsoldering from the hole you drilled. You can use a 'string thin piece' of electrical tape to wrap a figure 8 around both diode pins to secure it's positioning. You can also apply another tiny amount of epoxy in the figure 8 you made around the pins for extra securing. Once you think you've got a stable wrap around both diode pins, go ahead and try soldering a new lead on the replacement pin. If you have any methods of heatsinking the replacement pin without adding any stress or pressure to it, this is helpful too.
Every step I've mentioned above has to be done with extreme sensitivity and absolutely the least amount of pressure & stress on the new pin. This is not easy to do at all, you need to allow yourself a lot of time & patience to accomplish correctly reseating the new pin. As you can also guess, the diode connection you've made is extremely fragile. Even though it's soldered down, the bond is very weak so you cannot stress the pin at all when using the diode in your build.
I'm sorry for the long explanation, but I've painstakingly done this before & managed to salvage 2 broken pin diodes out of the 5 or 6 times this has happened. Good luck! :beer: