The stickies for replacing the cabin air filter showed how to remove the glove box. Once you have it outside in hands, shake it to let out the broken pieces of the latch. If something non-organic could drop in, it got to be able to come out the same way. And then you get 5 minutes to fix if using this
Super Strong Quick Drying Epoxy

 

The same thing happened to my latch a few weeks ago.

There’s no need to remove the glove box, but you will have to remove the screws from behind the glove box door including the handle, and then pry the back cover away from the front cover at the top.

The gap should be large enough for reattaching the latch (if it’s in one piece) with super glue at the broken mounting points, but be gentle with the door it’s going to be tight.

After you’re done just reapply some pressure on the glove box door and the two half’s will snap back into place.

Remember it’s only plastic so don’t force it too much.

 

Same problem with my '98 E320. Interior locking mechanism was no longer flush with the interior plastic glove box cover. RubenCR is correct. You don't need to pull out the box. After removing three screws, two that connect glove box handle on left side and single sheet metal screw on right side, I pried separation between the two panels with a flat head screwdriver. Start on the right side. In middle of box you'll then separate panels held together by two interlocking clips with gentle but firm persuasion. Lock had popped off two plastic posts holding it in place. Lips of plastic posts had worn. Plenty of room to simply reconnect lock to posts. Applied drop of crazy glue on each post, waited overnight, than applied a second drop. Before reassembling, test functionality of the locking box, making sure the spring on the assembly is engaged. The locking lever on the left should spring back up after being manually depressed. If it doesn't, the small protruding top portion of the spring needs to be pulled slightly to the right and clipped onto a notch on the plastic lock assembly.