  We detail numerical methods to compute the geometry of static vacuum black
holes in 6 dimensional gravity compactified on a circle. We calculate
properties of these Kaluza-Klein black holes for varying mass, while keeping
the asymptotic compactification radius fixed. For increasing mass the horizon
deforms to a prolate ellipsoid, and the geometry near the horizon and axis
decompactifies. We are able to find solutions with horizon radii approximately
equal to the asymptotic compactification radius. Having chosen 6-dimensions, we
may compare these solutions to the non-uniform strings compactified on the same
radius of circle found in previous numerical work. We find the black holes
achieve larger masses and horizon volumes than the most non-uniform strings.
This sheds doubt on whether these solution branches can merge via a topology
changing solution. Further work is required to resolve whether there is a
maximum mass for the black holes, or whether the mass can become arbitrarily
large.
