After several debates between the two of us as to the best way to replace the broken door, we decided we wanted two thin doors that opened out into the room instead of the old bi-fold version. Unfortunately, no one makes 12 inch doors. So, instead, we had to purchase a bi-fold door, take off the hinges and make it into two separate doors.
Unfortunately, however, because bi-fold doors have the special hinge and rail system, a door that is for a 24 inch opening only covers 23 inches. So we had a huge gap between our doors. Duff ingeniously fixed this by adding a nice piece of molding down the middle, which also helps make the door look like a unified piece. Overall, after our several dozen improvisations to make it work, the new closet door turned out wonderfully.
Here is a not-great before picture from when we toured the house (the bi-fold door is open in this picture):
And here is the after:
Good work, Duff!