First make sure you do see what you want. Make a schedule of museusm and sites and stick to it! The danger too often is that people on extended stays tend to put off sites because they get caught up in the day to day, always thinking they have enough time. Next thing you know you're leaving.
The next step is make the city smaller, view it in a way differently than a big tourist site that needs to conqured. Like most cities it is made up of little neighborhoods. WALK everywhere that you can, if you are able. Very often I randomly pick two things just for the walk between them. Don't try to make it interesting, that will come. This is a great way to explore Paris, and see the details of the city that you would miss.
Google for "unusual museums in Paris" or "free concerts in Paris" or once you are there pick up a magazine. THERE IS SO MUCH FREE STUFF IN PARIS, it is boggling, from concerts, to plays, to music, to reading. Try to find an event that is on the other side of town. The smaller the event the more "local" it will be. Go to Shakespeare and Co. (overpriced books) but they have a huge wall posting weekly events for expatriots, you have a chance to meet other Americans, living in Paris.
Don't forget to do nothing! One of the greatest pleasures of Paris is just living there, doing common things like laundry, eating cereal in you appt, buying milk and stamps, etc...Taking a good book down to the closest Jardin, and just stare at the sky while you pretend to read, all the while surrounded by....Paris.