A further option for finding a midpoint of coordinates would be a minimising function that finds the point with minimum great circle distance from each point.
Finding a more accurate method that is a) not affected by the choice of prime meridian and b) potentially allows for factoring in height values, brings up the idea of converting the coordinates from the KML file into cartesian coordinates with the origin in the centre of the earth. Averaging an reconverting those coordinates should give a more accurate centre-point within the earth from which a point on the surface could be approximated.