Constant light should be fine though. Flash is considered better because it lets you take shots instantly, but if you use the constant light which you already have, with a good tripod under the camera (and on a stable floor surface - avoid things like vinal or any surface that shifts with your own movements) and then just use a slow shutter speed and you should be fine (though make sure the watch is off otherwise hand motions will blur.
Settings tips: (check manual for more details)
1) Set camera to aperture priority mode
2) Set the aperture to f8 You can experiment with this, the bigger the number the more depth (area of photo in focus) you will get, however the longer the exposure will be because the bigger numbers result in less light passing through the lens. Note also that beyond around f18 the sharpness will deteriorate if you use bigger f numbers (smaller apertures) because of the diffraction effect.
3) Set the ISO to 100 or 200 - depends on the base ISO of the camera (manual should state this, but give the model and someone here should be able to chip in with the base ISO details as this might not be readily shown in the manual).
4) The camera will set the shutter speed based on the two settings above and on the light that you have. If you use the 2 timer or a remote to take the shot that should further remove any shake effect.
5) With the setup on a tripod the long exposure won't matter because nothing will be moving to cause blur. Note leave long exposure noise reduction OFF - this is noise reduction intended for exposures that last more than several minutes, which you should not be likely getting.