The 4960X is a very powerful, slightly older processor, but it is not especially fast. It is a very high quality 'big-rig' engine. Gaming works best with a 'race-car' engine, and of the choices you offered, the 6700K is the best for gaming.
However, it depends a lot on your motherboard.
The 4960X needs a X79 LGA2011 socket
The 5820K needs a X99 LGA2011-3 socket
The 6700K needs a Z170 LGA 1151 socket
If video editing, rendering, or recording and streaming is an important task, or if you use your system to make money, then the i7 5820K is likely the best choice. This it quite fast and very powerful, between the two other processors, but very 'general-purpose'