If you order something from amazon and it arrives a few days late, that doesn't mean you get the book for free. You could return the book to Amazon, but in your case your designs are custom made, so they can only 'return' them if they are obviously not as promised.
They are only entitled to a discount if you agreed on a deadline and you explicitly agreed that your payment was dependent on making that deadline. Or if they explicitly told you to stop working because they had decided to hire someone else, they don't have to pay you for whatever work you did after they told you to stop.
You could argue morally that if you promised to have it all ready in one week, but took three due to hanging out at the beach for two weeks, and that delay cost them money, then you might give them a discount.
But as far as the legal side goes, all a debt collector needs is confirmation that they ordered something from you, that you delivered it, and that they haven't paid what you invoiced them.