For the theoretical assessment of the $2^{3}P$ helium fine structure to become comparable to the precision measurements that have been made, it is necessary that the theory be calculated through order ${\ensuremath{\alpha}}^{6}m{c}^{2}$. In particular, the second-order contribution from the Breit and mass-polarization operators must be evaluated to an accuracy of 1% or so. In this work, for each of five possible intermediate state symmetries the Dalgarno-Lewis method is used to obtain the first-order perturbed wave function, from which the second-order energy follows by integration. Both the perturbed and unperturbed wave functions are expanded in Hylleraas-type series with a progressively larger number of terms, the second-order energies being computed at each stage; up to 455 terms are used for $^{3}P$ intermediate states and up to 286 for $^{1}P$, $^{3}D$, $^{1}D$, and $^{3}F$. The sequence of second-order energy results for each symmetry is extrapolated to the limit of an infinite number of basis functions to arrive at a final result. The $^{3}P$, $^{1}P$, and $^{3}D$ states will contribute to both the larger and the smaller fine-structure intervals ${\ensuremath{\nu}}_{01}$ and ${\ensuremath{\nu}}_{12}$, respectively, while $^{3}F$ and $^{1}D$ states affect only ${\ensuremath{\nu}}_{12}$. The total theoretical result, up to order ${\ensuremath{\alpha}}^{6}m{c}^{2}$, for ${\ensuremath{\nu}}_{01}$ is much more accurate than that for ${\ensuremath{\nu}}_{12}$, allowing the fine-structure constant $\ensuremath{\alpha}$ to be determined very precisely by comparison of theory to experiment, with the result ${\ensuremath{\alpha}}^{\ensuremath{-}1}=137.03608(13)$, good to 0.94 ppm.