My glove box refused to open a number of years ago.  I was able to open the box door without damage. You will need a flashlight to see under the latch handle and a very small screwdriver. The kind you fix eye glasses. It helps if you recline the passenger seat and move it forward so you can rest your back on the seat while getting close to the glove box lock.

The following instructions could result in damage to the glove box door.

Pull the handle out, don't worry about breaking it because it already is, till you see a little silver cylinder. With the screwdriver, try to rotate the cylinder toward the dash. You might "play" with that several minutes before you "hit" it just right. If you were as lucky as I was, the door will just pop open.

The handle is toast and will have to be replaced. The problem, these latches should never have been made of plastic. They were pretty easy to break when they were new. Now, more that a decade old, they bite the dust very easily. I was lucky to find some new ones a few years ago.

To add to Howards solutionbelow,  is a sketch showing the metal part that holds the latch.

When you pull out on the handle, it pushes down on the metal part and that releases the latch (assuming it is not locked)

As Howard indicated... lift the latch handle and you will see the silver metal part, with whatever you have in your toolbox, pull the metal part down.

The common failure mode is the plastic part of the handle that pushes down on the metal part breaks.

Barney Eaton