I would first design a great contact form--make sure you follow best UI/UX practices.
But now, although you have designed a greater looking page and greatly improved the UX, the numbers are still the same, no one is signing up. What do you do? Even Google Analytics will never tell you why people aren't signing up. They'll tell you what, where, when, and who, but never why. For that, you need something that gathers user surveys. You can use something like Qualaroo that allows you to target certain groups of audiences and gain their insights. For example, for every user that goes to your About Us page and spends at least 1 minute but doesn't choose to sign up, a survey will pop up and asking questions, "How can I help you?" "What are you looking for?" That will get you the qualitative data you need to know why your bounce rate is # or why your page viewing times are #. There was another tool called Crazy Egg which could record every users visit that didn't sign up but that would entail going through every video of every user that didn't sign up. And if your conversion rate isn't high, that will be a long task.
Essentially, I'd tell your employer that you can redesign the page to make it as user friendly and visually appealing as possible, but there's no magic sauce from your end that will instantly increase conversions in this case. Perhaps your company needs to have a bigger discussion about marketing efforts, you can recommend for them to create a blog and include a sign up form there as well, perhaps more engaging content on their blog will gives users a bigger incentive to sign up.