Quote by PeterDonis
the 2sphere at the EH, where r = 2M in Schwarzschild coordinates, is not spacelike; it's null. So it is physically impossible to perform the comparison I described using a 2sphere at the EH, and there is therefore no way to physically define the K factor (or the J factor, for that matter) at the EH.

On rereading, I should restate this. The 2sphere at the EH can still be said to have a physical "area", which is 4M^2 in geometric units. So it may not be technically correct to say the 2sphere itself is null. (When I compute the norms of the tangential unit vectors, I don't get zero at r = 2M; the norms are still positive, so the unit vectors are still spacelike, assuming I'm doing the computation right).
However, there can't be a static 2sphere "hovering" at the EH, because the EH, as a surface in spacetime, is a null surface, and therefore there can't be a surface of "constant time" that is orthogonal to the EH, in which the 2sphere could be said to lie, and in which the area of the 2sphere could be compared with the volume between it and a neighboring 2sphere. So the main point in what I said above still holds: it's impossible to do the physical measurement at the EH that I was using to define the K factor.
I should also note that the above does not entirely apply to the J factor; since there are still timelike worldlines passing through the EH, it is still possible to define a "gravitational redshift" factor there, for an infalling observer. However, this factor cannot apply to a static, "hovering" observer at the EH, since as we've seen there can't be one. So what I said does apply to the J factor for static observers.