We present a high-order Nystr\"om method for Neumann boundary value problems on domains with corners. Key difficulties which must be addressed in the solution of the associated (second-kind) integral equation are related to the fact that solutions themselves are unbounded. This not only makes integration difficult, but also gives rise to (potentially enormous) subtractive cancellation. We overcome these difficulties by exploiting knowledge of the asymptotic behavior of the integral equation solution, thereby providing a high-order method. Numerical experiments demonstrate its efficacy on a variety of domains, including those with corner angles as small as \pi/100 and as large as 199\pi/100.