We are concerned with a 2D time harmonic wave propagation problem in a medium including a thin slot whose thickness $\epsilon$ is small with respect to the wavelength. In Part I [P. Joly and S. Tordeux, Multiscale Model. Simul. 5 (2006), no. 1, 304--336 (electronic); MR2221320 (2007e:35041)], we derived formally an asymptotic expansion of the solution with respect to $\epsilon$ using the method of matched asymptotic expansions. We also proved the existence and uniqueness of the terms of the asymptotics. In this paper, we complete the mathematical justification of our work by deriving optimal error estimates between the exact solutions and truncated expansions at any order.