  We show uniqueness theorems for Kaluza-Klein black holes in the bosonic
sector of five-dimensional minimal supergravity. More precisely, under the
assumptions of the existence of two commuting axial isometries and a
non-degenerate connected event horizon of the cross section topology S^3, or
lens space, we prove that a stationary charged rotating Kaluza-Klein black hole
in five-dimensional minimal supergravity is uniquely characterized by its mass,
two independent angular momenta, electric charge, magnetic flux and nut charge,
provided that there does not exist any nuts in the domain of outer
communication. We also show that under the assumptions of the same symmetry,
same asymptotics and the horizon cross section of S^1\times S^2, a black ring
within the same theory---if exists---is uniquely determined by its dipole
charge and rod structure besides the charges and magnetic flux.
