Try setting the drive mode to self-timer (delay), single frame, continuous - all the different settings. If this doesn't work, use RESET from the set-up menus; this is always worth trying. Always try a fresh memory card, any time there is a problem. Finally, it's worth removing the battery entirely and leaving the camera for at least twenty minutes, then replacing the battery. If none of this works, they shutter release may have a problem with the second stage of contact (the first stage does the focusing). I don't know if the 5200 has a socket for a remote release or cord release. That can be way to test if the shutter release itself is OK. Servicing can cure the problem...
(A mate wrote that for me. I thought they'd be able to help)