Just get lost. There's no better way to experience all that is Paris. Wear comfortable walking shoes (the only un-Parisian element of this intentionally-vague itinerary) and spend hours making your way in and out of the metro, up and down from street to train, under and over the Seine and back again. There's a romance and a freedom to it. Do not over think it.
I recently flew through Paris on my way home from a vacation. When booking my flight I debated if I'd be up for one more city. In the end, adventure won. It always wins. I booked the longest layover I could (19 hours), and rode the RER-B train into the Saint-Michel metro station in the Latin Quarter. I dropped my bag at the hotel and wasted no time getting lost.
The magic of Paris is in the details. You can only feel Paris as you walk right through her. You smell the sweet boulangeries, you see every flourish of Beaux Arts, you hear one of the sexiest languages to ever touch the human tongue rolling and rolling among friends at cafes. The magic is tangible. Paris is a sensory feast and to share that moment in time with the locals you have to ditch the maps.
My favorite memory of Paris is leaving Notre Dame on the evening of Christmas Day, walking aimlessly in the opposite direction of my hotel, and stumbling onto a trio of jazz musicians. The moon was full, the river flowed quietly beneath us, the air was crisp, and people danced joyfully with complete strangers.