Thanks for asking the first question that forced me to pick a paper and pencil. Assuming K can be any thing in range [0, infinity] and that all sub-circles must have equal diameter, I suggest the the following solution (which will most likely be wrong): From the center of the circle draw K radial lines 360/K degrees separated from each other. In each pizza slice draw a circle tangential to the two radial lines defining the slice and the periphery of the original cericle.
For K = 1, this will yield the orginal circle. For K = infinity it will yield very tiny circles sticking to the ourter boundary of the original circle with tons of empty space. For everything else, it will be something in between. Once you do the math it will be O(K). Some circles will be undrawable (e.g 360/13) is probably not a rational number. Sorry if I am wrong.