The distance is strictly greater than $2$ --- as is true for any compact, nonhomeomorphic $K_1$ and $K_2$ when $K_1$ is totally disconnected (Cohen-Chu, Stud. Math. 1995, p.6).  The question of an upper bound is more puzzling.  @HannesThiel comment 6/13/2016 suggests distance $\le 16$.