The Gordian distance between two knots is the fewest number of crossing changes necessary to transform one knot into the other. Khovanov homology is a categorification of the Jones polynomial that comes equipped with several spectral sequences. In this talk, we show that the page at which some of these spectral sequences collapse gives a lower bound on the Gordian distance between a given knot and the set of alternating knots (and also on other related Gordian distances). We also discuss connections to the existence of torsion in Khovanov homology.