Imagine we have a king by itself on a chess board, making random moves around the board. Although it is apparently aperiodic, wouldn't the corresponding Markov chain to the King's movements be periodic since the King could only return to a square i on the board at moves 2,4,6 ... etc. (great common denominator of 2) after initially being at the square?
By definition, aperiodic chains have return times to i with a g.c.d. of 1.
Your mathematics is correct but your understanding of the rules of chess is not. A king on any square that is not on the edge of the board can move to any of eight squares: the four that share a common edge with the one he's on and the four that share only a common vertex.