Why would back sliding doors not latch shut?

It could be that the latching mechanism is either stuck or not performing correctly. If you have some pictures, that would help. With the handle being broken off, it may be that the latching mechanism is engaged like you are opening the door and it cannot return to the locked position to close the latch. It’s hard to see without seeing it in person. For safety and the quickest remedy would be to take it to a body shop to have them repair it.