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When a high-frequency acoustic or electromagnetic wave is scattered by a surface with a conical point, the component of the asymptotics of the scattered wave corresponding to diffraction by the conical point can be represented as an asymptotic expansion, valid as the wave number $k \rightarrow \infty$. The diffraction coefficient is the coefficient of the principal term in this expansion and is of fundamental interest in high-frequency scattering. It can be computed by solving a family of homogeneous boundary value problems for the Laplace--Beltrami--Helmholtz equation (parametrized by a complex wave number--like parameter $\nu$) on a portion of the unit sphere bounded by a simple closed contour $\ell$, and then integrating the resulting solutions with respect to $\nu$. In this paper we give the numerical analysis of a method for carrying out this computation (in the case of acoustic waves) via the boundary integral method applied on $\ell$, emphasizing the practically important case when the conical scatterer has lateral edges. The theory depends on an analysis of the integral equation on $\ell$, which shows its relation to the corresponding integral equation for the planar Helmholtz equation. This allows us to prove optimal convergence for piecewise polynomial collocation methods of arbitrary order. We also discuss efficient quadrature techniques for assembling the boundary element matrices. We illustrate the theory with computations on the classical canonical open problem of a trihedral cone. [ABSTRACT FROM AUTHOR]