Sure, the 60D can shoot video, and the 17-40mm lens mounts on the 60D. There's no reason why this combo won’t work. I don’t shoot much video, but when I do, I tend to prefer lenses with image stabilization. The 17-40mm doesn’t have IS, which may or may not be an issue for you; depends on what you’re shooting and how you’re shooting it.
You pose a difficult question to answer because it’s so general.
Does your body/lens combination work for video? Yes. Is it suitable for shooting a wedding video? Sure, if that’s the focal range you’re after.
That said, I feel like the general consensus is that the 24-105mm is the better lens for video due to the 3-stop IS and the more versatile focal range. If wide-angle video is what you’re looking to shoot, then of course, the 17-40mm is more suitable.
Since there’s no IS, try to stabilize the camera (e.g. steadicam, rails, etc) or at least use a tensioned strap. On a full-frame body, any camera shake may be inconsequential due to the wide angle view, but with the crop factor of an APS-C body, it’s more apparent.
Comments
I don’t shoot much video, but when I do, I tend to prefer lenses with image stabilization. The 17-40mm doesn’t have IS, which may or may not be an issue for you; depends on what you’re shooting and how you’re shooting it.
Does your body/lens combination work for video? Yes.
Is it suitable for shooting a wedding video? Sure, if that’s the focal range you’re after.
That said, I feel like the general consensus is that the 24-105mm is the better lens for video due to the 3-stop IS and the more versatile focal range.
If wide-angle video is what you’re looking to shoot, then of course, the 17-40mm is more suitable.
Since there’s no IS, try to stabilize the camera (e.g. steadicam, rails, etc) or at least use a tensioned strap. On a full-frame body, any camera shake may be inconsequential due to the wide angle view, but with the crop factor of an APS-C body, it’s more apparent.