The reason you run three wires (two hots and a neutral) and ground to a range or dryer is because those appliances have controls that are just 120v whereas the heating elements are 240v. If the kiln has just heating elements and one leg (or both)of the 240 is switched thermostatically, then all you need is a two pole breaker with two conducting wires and a ground going to the kiln. 50 amps requires that the service be at least 6 AWG copper. When you say the breaker you bought is 60 amp, then you have to run 4 AWG. The breaker is sized to protect the wire connected to it. The wire and any receptacles in the circuit are sized for the load being supplied. Go back and get a 50 amp two pole.Run 6/2 w/GND AWG copper. The number ten in your kiln is smaller because wire size rules are different for appliances than for your home and different types of insulation are used.
Go on the WEB and search Google for more information on your kiln. You can probably get a wiring diagram for it off the net. If you are still uncertain what you have, post back or call an electrician. It's not rocket science, but you do want to be sure what you are doing with electrical. Luck, Kizmet