The button for the glove box, the button for the eyeglass compartment, and the lock between them are a single assembly. It can be removed so that you can get to the cables that operate the glove box and the eyeglass compartment latches.
Put your ignition key into the glove box lock slot. Turn the key 45 degrees to the right. Insert a small diameter (like a straightened paper clip) wire into the edge of the lock cylinder where the small hole is. Push in with the wire while pulling on the lock a bit. You need to release the clip at the bottom of the cylinder. Its tricky, but the cylinder will come straight out. Wiggle, press, curse, but don't give up.
Once its out, do NOT remove the key... the cylinder pins will fall out.
Once the lock is out, remove the screw at the bottom of the lock hole and the whole assembly will come out of the dash.
Sounds like the cable is disconnected or misadjusted. Pull on the cable wire and the glove box will open. Once the box is open, you can remove the box liner if needed and you'll see how simple the latch system is.

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KenP

1996 S500 W140 Coupe