If you have a cable release this will be easy. If not, it's a bit trickier, but still possible.
Cable Release Jack
On the side of your camera you should have a cable release. On more entry-level models, it should be a 2.5mm jack, like this one on my T5
A Cable! My Kingdom for a Cable!
If you have older video out cables from some point-and-shoot cameras or some video accessories, you may find that they used a 2.5mm jack. At least that's where this one came from. The RCA side (or not) isn't terribly important - the size of the stereo plug on the other side is. I believe it's also important to have the three bands vs. two. It should fit into the cable release jack on the side of your camera.
If you have RCA ends like mine, then do it this way. If you have another stereo jack on the other end you'll need a paper clip, a penny, or some other conductive material. You just need to short (i.e. connect) the right two bits. The cable release jack has 3 connections in it - ground, half-press, and full-press. In my case, I have 4 connection points on the RCA sides. Two posts, and two sheaths. That means I can connect post-A to post-B, post-A to sheath-B, sheath-A to post-B, and sheath-A to sheath-B.
One of these things should work for you. Maybe.
Why This Might Work
If this works, you have something horked up with your shutter button. That seems unlikely to me, but if you have a cable around like this it's the easiest thing to check.