  We produce new examples, both explicit and analytical, of bi-axisymmetric
stationary vacuum black holes in 5 dimensions. A novel feature of these
solutions is that they are asymptotically locally Euclidean in which spatial
cross-sections at infinity have lens space $L(p,q)$ topology, or asymptotically
Kaluza-Klein so that spatial cross-sections at infinity are topologically
$S^1\times S^2$. These are nondegenerate black holes of cohomogeneity 2, with
any number of horizon components, where the horizon cross-section topology is
any one of the three admissible types: $S^3$, $S^1\times S^2$, or $L(p,q)$.
Uniqueness of these solutions is also established. Our method is to solve the
relevant harmonic map problem with prescribed singularities, having target
symmetric space $SL(3,\mathbb{R})/SO(3)$. In addition, we analyze the
possibility of conical singularities and find a large family for which
geometric regularity is guaranteed.
