It could just be that the button has failed.
Or that the black and white wires are swapped.
I would recommend getting an outlet tester that has a GFCI test button on it. It makes checking for correctly wired outlets much faster. It is also the correct way to test that the GFCI works for downstream outlets.
There should be power between the ground and the narrow side.
(If it is the wide one it is installed wrong, and the button can't work.)
If that button on the tester also doesn't trip it, you either have a failed GFCI or one that is installed wrong.
There are two sets of terminals on a typical GFCI.
One labelled LINE and the other one LOAD.
You have to make sure that the power is connected to the two terminals that are labelled with LINE.
Any other outlets that are protected by the GFCI would be connected to the terminals labelled LOAD. (There often isn't anything connected here.)
If you connect it backwards it won't matter if it is tripped or not as the power would be directly connected to the outlet. (There are some newer ones where you would have the opposite problem instead.)
Assuming you don't know which cable is which here is some directions. If you don't feel comfortable doing this, don't do it.
If it had been working until now, just replace it.
(In which case many of these steps can be skipped as you just connect the wires on the old LINE to the new LINE. )
These are the steps I would do if I couldn't otherwise figure out what was wrong.
- shut off the power
- check that it is off at the GFCI
- take it out and disconnect all of the wires
- make sure the wires aren't touching anything.
- keep everyone and everything away from the wires.
- turn on the power
- carefully use the multimeter to determine which cable has power
(don't touch anything with your fingers except the multimeter)
- shut off the power
- check that it is off
- connect the power cable to the LINE terminals
(black to the brass colored terminal, and white to the silver colored terminal)
- connect anything else that needs protection to the LOAD terminals
(often there is nothing connected to them)
- install the outlet
- double check everything
- turn on the power.
- test that it works correctly
If it still doesn't work correctly there may be a problem with the GFCI itself. In which case you probably need to replace it.
(I have had to replace a brand new GFCI because a little plastic piece inside it broke.)
If the existing one is discolored or dirty, I would just replace it.
(I might still follow the steps above if there is any question about it being wired correctly.)