I think that if you're really careful and set up a bit of a jig, you could solder your broken pin piece back on.

One idea would be to poke your pin piece slightly through a piece of masking tape, then stick the tape to the ends of the other pins, so that the broken pin is lined up where it is supposed to go, on its own. Then solder the pin to repair it and scrape away excess solder carefully with a sharp razor blade.

There is a risk with this method that the pin can break off more easily potentially leaving it stuck in a socket, so this might not be the best solution if you wish to remove / reconnect this thing a lot of times.