Pinoy, all of the big players in control make something that will work for you ~ Eurotherm, Watlow, Omron, you name it. Rashid nailed it with what he recommended ~ just look for one that is set up to use standard 120 vac and you are good to go. As Rashid stated, they can all drive a SSR (solid state relay) that in turn will turn off and on the heating element, and they all use K type TCs as one of their options. The controller will not directly switch the power, so rating of the contacts in it are not important. Make sure that the TC is rated for the temp and atmosphere that you want to achieve. As far as your question about accuracy, any of the quality ones are more than accurate enough. Also, most of the newer units are set up to "auto tune". This allows the controller to teach itself how to best reach the set point temperature without overshooting, and also allows it to stay at steady state temperature without fluctuating much. Sometimes it still won't be perfect, and you will need to manually alter the PID settings some to get it just right. Our equipment that we build to use in our factory uses something called a phase angle firing current controller(which is just a fancy adjustable thyristor) instead of the SSR. It gives us even better control of the heat and the elements last longer.
Best wishes with your project,
Steve