Are you planning to run this kiln on an extension cord or hard wired in? In either case, you should always stick to what the appliance is rated at. If it requires 10 gauge wire, you need to use 10 gauge wire or heavier, but not lighter. Even though you say this kiln is small, you can still start a fire with under rated wiring. The wire can over heat. It is best to stick with the wire that the kiln was rated for. Since it draws 15 amps, most home circuits can handle that. Many appliances, like toaster ovens, draw that much current. However, your kiln will be on and drawing power for a longer period of time, which can cause over heating of under rated wiring, especially if there are any other appliances using the same circuit. With an appliance such as a kiln, I would never use an extension cord. However if you must, make sure the cord is rated higher than what the appliance requires. In this case, I would use 8 gauge or even higher. Never run the appliance off a long extension cord. I am not an electrician, but I do have experience in this field.
Just a thought, if your garage has power, it should have it's own breaker and circuit. Check the breaker box in your home, to see if the garage has it's own circuit. I have a detached garage as well that has full power. The garage has it's own breaker box. In my previous home however, the garage had power, but no breaker box inside.