A cosy café in central Paris that is open late at night?

Arguably... there is no single place that can meet all of your requirements and you will be making trade-offs among the various candidates. Acting on the assumption that the primary requirement is that the establishment stays open until 1 or 2 AM, and secondarily that the ambiance is amenable to a romantic encounter, there are several interesting candidates...

Cosmopolitan

Le Fumoir is located at 6 Rue de l'Amiral de Coligny, a stone's throw from the Louvre. There's a "back room" for more a intimate ambiance. It's open until 2 AM.

Comfortably Geek

Dernier Bar avant la Fin du Monde is located near Châtelet and offers plush armchairs situated around coffee tables. It has a library-style décor in one part and exhibitions by aspiring artists in another; it's open until 1 AM seven days a week.

Belle Époque

Le Zimmer located at 1, place du Chatêlet is open until 2 AM except Sunday when it's open until 1 AM. You can sit outside and watch the world go by or take a seat inside and enjoy the elegance.

Eternal Paris

In this option you act as the avant garde beforehand and scout out a locale. To do this, follow the Rue Moufftard north to Place de la Contrescarpe. In this neighbourhood you will see literally dozens of cafes meeting most of your requirements. As budget is no object, you may be able to slip the proprietor several hundred Euro thereby convincing him/her to remain open until you leave. Also if you are seated outside, you will be able to hire a street musician to play your favourite songs.

Source: Google Street View, fair use

Note: to check things out, I used the search term: "gemütlich cafe paris". It turns up lots of results, but the overwhelming majority close at 7 or 8 PM.

Note: about requesting your favourite song, some of the more up-to-date cafes are installing the new Yamaha sound system that lets you request a song from your mobile. I haven't tried this yet but the specs sound fascinating.