Deb, how many nights in Sicily? At what time is your flight out from Catania on your last day?
Picking one home base for all that you want to cover will not be possible, without spending your entire days in the car. You may want to choose the last 2 nights either in Siracusa (Ortigia) or Taormina. Your first night could be somewhere like Erice, Agrigento or Piazza Armerina, for example. Maybe consider flying from Rome into Trapani.
One thing for sure, you will have to narrow down what you will be able to realistically see from your list. Eliminate Enna right off the bat - the A19 is undergoing under major repairs in the coming months, with one part closed. Only combine two places that are close by to each other, for each day's plan. Count on delays: slow driving, slow meal times, etc.. You will love Sicily and kick yourselves for not having allocated more time!
IMO, Agrigento, Siracusa and Taormina are the top three to see in a very short time. Although I really like the areas around Palermo and the interior, you don't have the time. Siracusa can be your base for the duration if your plan sticks to these top 3. Between Agrigento and Siracusa, you could take the coast road, with a stop in Noto, for example.