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).  More puzzling is a specific upper bound.  None seems to have yet been offered here? None published?  Is the distance greater than, say, $1000$?