Inspiring Quilters, Stitch by Stitch
It is very likely that the original binding did not have mitered corners. The quilter probably used a bias binding and just "curved" it around the corners. So your option is to either round the corners again or remove the original binding and miter the new binding at the corner. I would base my final decision on how to handle the binding on what the owner's plan for the quilt is. If it an heirloom then I would remove the old binding and start fresh. If the desire is just to preserve it for future use go on with your original plan and curve the binding around the corners.