On my Kodak C813 the shutter button stays on record, when I turn the button it will not work on taking picture or anything else, but record.

It seems that there is a problem with the rotary mode selector switch. Could be dirt or a faulty contact. Self repair is inadvised. Might be cheaper to replace the unit, and keep the C813 as a still only camera.

It sounds like it isn't going into record mode.
Can you get into the main menu?

Is your lens clean? Are you pressing the shutter button half-way and allowing the camera to lock in the focus before pressing the rest of the way? Is the lens coming out correctly?

Make sure that you're not using 'manual' mode and too short exposure/too small aperture. Check that your flash is coming to play also.

A stuck shutter is another common failure mode for digital cameras. The symptoms of a stuck or "sticky" shutter are very similar to CCD image sensor failure. The camera may take black pictures (for shutter stuck closed), or the pictures may be very bright and overexposed, sometimes with lines, especially when taken outdoors (for shutter stuck open). To confirm a stuck shutter, put the camera in any mode other than "Auto", and turn the flash OFF (you don't want to blind yourself for the next step). Next look down the lens and take a picture. You should see a tiny flicker in the center of the lens as the shutter opens and closes. If no movement is seen, then you likely have a stuck shutter. If so, please see this link for further info and a simple fix that may help.

Kodak Easyshare C813 can only take picture, transferred pictures and video to your computer and record video. C813 cannot be use as a webcam if you'll like a kodak webcam you can check this products like Kodak S100 webcam, Kodak W100 webcam and T130 webcam. Check the website www.kodak.com/go/shop and for information about your camera you can visit www.kodak.com/go/C813.

you need bring it to the service center of kodak to check

Maybe the lens is intact (stuck).

with kodak easyshare c813 light not available in mode taking video

Preliminary searching leads me to believe that this camera won't take anything larger than a 1GB memory card.