I'm pretty sure that the button is not making contact to trip the shutter. In a basic release they have a two stage button, and the first stage closes to start AF/Metering, and the second stage fires the shutter. Internally in the camera the remote release switch just seems to be wired in parallel to the actual shutter button.
In an emergency you could just cut the switch off the wire, strip back some of the wire and try touching them together. One of the three wires will be common, and the other two will be AF and Shutter. Since usually all three are connected together when the shutter fires you have no danger of damaging the camera trying different combinations to find the correct function of each wire. The common is most likely to have black insulation, but you might not have a black wire, and it may not be connected as the common, it would just be a starting point for me. You only have six options, including none and all three connected, so it won't take long to figure out. This Macgyvering will get you out of trouble, at least on a landscape or other shoot where the subject is not moving about too fast.
For the more advanced types of release, with the built in intervalometer, they have far more possible points of failure. Any of the components on the internal PCB could fail and cause partial or total failure of the unit. Depending on where the failure has occurred it may well be possible that you might be able to set the intervalometer and that will actually trigger the camera, even though you can't do it directly via the button. Even if you put a new battery in it I would still try another one, from a different source, it is not unheard of to get a duff battery after all.
Alan