  We compute explicitly the first-order in $\alpha'$ corrections to a family of
solutions of the Heterotic Superstring effective action that describes
fundamental strings with momentum along themselves, parallel to solitonic
5-branes with Kaluza-Klein monopoles (Gibbons-Hawking metrics) in their
transverse space. These solutions correspond to 4-charge extremal black holes
in 4 dimensions upon dimensional reduction on $\mathrm{T}^{6}$. We show that
some of the $\alpha'$ corrections can be cancelled by introducing solitonic
$\mathrm{SU}(2)\times \mathrm{SU}(2)$ Yang-Mills fields, and that this family
of $\alpha'$-corrected solutions is invariant under $\alpha'$-corrected
T-duality transformations. We study in detail the mechanism that allows us to
compute explicitly these $\alpha'$ corrections for the ansatz considered here,
based on a generalization of the 't Hooft ansatz to hyperK\"ahler spaces.
