We study the scattering of a polarizable atom by a conducting cylindrical wire with incoming boundary conditions, that is, total absorption, near the surface of the wire. Based on the explicit expression given recently [C. Eberlein and R. Zietal, Phys. Rev. A 75, 032516 (2007)] for the nonretarded atom-wire potential, we formulate a hierarchy of approximations that enables the numerical determination of this potential to any desired accuracy as economically as possible. We calculate the complex $s$-wave scattering length for the effectively two-dimensional atom-wire scattering problem. The scattering length $\mathfrak{a}$ depends on the radius $R$ of the wire and a characteristic length $\ensuremath{\beta}$ related to the polarizability of the atom via a simple scaling relation, $\mathfrak{a}=R \stackrel{~}{\mathfrak{a}}(\ensuremath{\beta}/R)$. The ``scaled scattering length'' $\stackrel{~}{\mathfrak{a}}$ tends to unity in the thick-wire limit $\ensuremath{\beta}/R\ensuremath{\rightarrow}0$, and it grows almost proportional to $1/R$ in the opposite thin-wire limit.