How does a professional go about ensuring that their clients needs as well as the employer's demands are all successfully met simultaneously?

I think you need the right information and correct tools, information can be used as recording system of customers and employees as survey apllication. And for the tools can be described as strategies, satisfaction for customers, motivation for employers or process that being used to the final task

For customers you can make a survey about the service they receive, satisfaction, advice, compline and overall rate

But for employers I describe it as the horizontal strategy that try to build good relationships with the employers if they are few but if they are many you can let someone to get their reasonable demands from time to time and take their advices that can let them they are part of the organization and motivate them in a way . this is my theory or what i think of.