It's tricky, but a key success metric is and always shall be one thing: money. Did the design, when implemented, save the company money (by reducing costs) or make the company money (by increasing sales of an item). I have found if you don't have "baseline" data to compare the new status quo you implement to the old then it's awfully difficult to prove the busines value (you just don't have the data).
I've been fortunate enough to design and implement new solutions that "moved the needle" significantly – basically, the new version of the design I created, when implemented, showed improvements in the key metrics that the business was tracking against (customer satisfaction, time-on-task, and adoption/uptake). In one instance I had a "hockey stick" uptick – the numbers jumped dramatically, and stayed there. Why? Because the design I was replacing had pronounced usability issues, and the new design I created was markedly better and also was easy to understand, use and (a key factor) RE-use.
One other key success factor was I knew the goals of the design GOING IN – understanding the business drivers helped me form a happy balance between the business and the customer, which in turn allowed for a more effective solution. If you create a great design that doesn't support the business, you may produce a solution that is absolutely great… but doesn't service the business's needs (which helps pay the bills, including the overhead that is, well… you).
Sherlock Holmes famously stated "I cannot make bricks without clay." Facts – analytics of the before and after – are absolutely vital to speaking to the business. It may be "soft" numbers, like customer satisfaction or "net promoter score" – but it's still data that you can use to show the value of UX design.