Looks pretty good to me. What sort of problems is it creating for you? Why do you need more accuracy? The distortion is probably computed as a regression through the detected corner points, and since it's a regression may not have subpixel accuracy for all points in the image. How much off is it and what problem does that cause in your analysis of the scene?
Your camera is also not pointed at the center of the scene and probably does not have the optic axis normal to the plane of the gray frame, thus causing skew/shear in the corrected gray frame. Is it super important that the gray frame be perfectly rectangular? If so, why? Try adjusting your camera so that the camera is at the center and perpendicular to the plane of the scene.
It might also improve if you made your checkerboard larger and put it near the center of the scene with its center on the optic axis.
It looks like you're using a very short focal length lens and are close to the scene, thus causing the fish eye (barrel) distortion. You could reduce the distortion in the first place by using a longer focal length lens, or course this would mean moving the camera farther away to get the same field of view, but maybe that's an option.
