Itinerary critique wanted!

Good luck with an Eiffel tower picnic at night in February. Because of French school holidays, even if you're making photo-op stops and not going in, you might find places more crowded than you intended.

I think this is doable; however, you need a Plan B for the picnic at the Eiffel Tower because in February it is going to be cold and possibly wet.

For your birthday brunch, I can recommend Les Ombres. It is at the Branly Museum and has an excellent view of the Eiffel Tower, looking up through an atrium. It is pricey, though. I had a 3-course lunch for something like 49€, but the real shock was that my glass of champagne was an additional 21€ - just be forewarned. I did enjoy the food and thought is was good value for the view and experience (despite the price of the champagne), and it was my one splurge lunch. Splurging at lunch is much less expensive than splurging at dinner.

Rather than tea at the Laduree on the Champes Elyses, go the the Laduree on Rue Royal, which runs between La Madeline and La Concorde. It is very elegant and less touristy than the one on the Champs. And, there is nothing special about the Champs Elyses other than the Arc du Triomphe - it is just a long street with lots of chain stores and hordes of tourists.

Bon voyage...and Happy Birthday.

That's great that you two will enjoy the time together.
I'm still unclear on "is there a better way to do it?"
Do you mean more efficiently do "it", or what other places should we consider, or both?
I do agree about skipping Versailles.

In any case, I've done this with each of my 3 kids when they were around 14.
One thing that you've perhaps already done is to have your child have some input.
Does she want to go back to her friends and say she's seen the Mona Lisa or Napoleon crowning himself emperor or had an unlimited bowl of chocolate mousse?
Here's what my daughter liked at age 13: the Sewer Tour, The Louvre, shopping in a Paris supermarket, riding the Metro, walking up the first flight of the Eiffel Tower, and The Catacombs. (We repeated the last two last year when she was twice that age.) We also enjoyed going to the cafe Chez Janou where one desert is ALL the chocolate mousse you can eat! It's off the beaten tourist path so it is very French, they don't speak English, but that's half the fun! (Put Google Translate app on your phone if that worries you.)
I am wondering why you don't want to go in the Louvre. She would be free, and Rick Steves has a nice 45 min tour that you can download for free on to your smart phone.
https://www.ricksteves.com/watch-read-listen/audio/audio-tours/paris

If your daughter might like some history, you can go on free tour near Notre Dame, which we liked and took about 90 min. You could even do that on the first day to sort of set the stage and get oriented: http://www.neweuropetours.eu/paris/en/home#
They also offer tours of Montemarte, which will make it more interesting than wandering around there on your own.

I know, I've added to your list of things to see, but if even one of them sounds good, you could take out the 2nd visit to Eiffel and fit other things in. And, if you are efficient and use the Metro, you can fit a lot into a day.
Also be flexible, so that you can see St. Chappelle or Sacre Cour on a sunny day, and inside sights on a cloudy day.

Finally, you both might be interested in seeing the Intro to "Midnight in Paris" which is free on youtube (4 min): https://www.youtube.com/watch?v=JABOZpoBYQE

Have fun,
Allen

Eiffel Tower is not surrounded by restaurants. But there places to the east on that bank of the river. If you call up Rue Clair on Google Maps, and click on Nearby, there is a function that will show restaurants, often including links to the establishments. It may not be totally thorough and offers no evaluation of quality but it is a good way to start your search.
PS: Even in summer the tower itself is not convenient for picnics; both there and the Trocadero are paved with concrete and offer little place to sit down. The great lawn in the other direction from the tower does not welcome picnics. Along the river bank you might, in season, find something suitable, i.e. seatable.

Wow! Thank you so much for all of these extremely helpful suggestions!! I really appreciate it. And will defintely be adding Les Ombres on my must do list!

I didn't think about the wet/cold aspect of the picnic! Gosh, that is one of the biggest things we wanted to do. Wondering if I shouldn't re-schedule the trip for April. I just love the idea of spending my ACTUAL birthday in Paris. Hmmm..... Already have plane tix.

Hi crispers, don't cancel your trip. Spending your birthday in Paris is a wonderful idea! You have 2 full days to soak up the atmosphere of this world-class city.

The 7th arrondissement where you're staying is my favorite. There are several cafes where you can have a meal and/or drink and catch up with your daughter. There are worse things than just hanging out in Paris! Try the Café du Champs du Mars and Alfio (it's a small, Italian place, great food). Both are near the Eiffel Tower but no actual view. Agree that Les Ombres is a good choice for a birthday dinner.

When you're exploring Rue Cler, walk a bit further to see Napoleon's Tomb. Add a visit to either the Louvre or Orsay.

Hope you have a great trip!

@Donna

Thank you so much for these suggestions!! I think we will stick to the plan for Feb...this just means I will have to go back in the spring sometime :)

Happy Birthday. My first thought, that's a lot of Eiffel Tower for not even wanting to go up. I'd suggest getting that down to only one visit (especially in Feb.) and focus more on indoor activities that time of year - Louvre, d'Orsay and Orangerie are just a few choices. Use Trip Advisor or guidebook and go down the list of possible things to do. I always list them as must-see, nice to do if I have time, and something I can skip. The ratings are based on my own interests. I plan two "must sees" a day, one in the morning, one in the afternoon with lunch in the middle. I add "good to see if I have time" when I can and add evening activities when I'm up for them.

I haven't been to Moulin Rouge yet but have heard it praised by locals. It might be a good visit during low season.

The cruise is better after dark, so pick a night that isn't miserable. It's best in open air viewing the lights. You won't need a reservation or to plan ahead. I've only been on Vedettes du Pont Neuf but there is one that leaves from the Eiffel Tower area too.

Saturday - You'll want more time for the morning plan. The Orsay will probably be very crowded - it usually is. Consider going to the Orangerie instead. Skip the Eiffel Tower. Take the Seine cruise with Les Vedettes du Pont Neuf (buy anytime tickets online before your trip). It's a short walk from the Sainte Chapelle. Hot chocolate and a pastry at Angelina's is lunch!

Sunday - last place I want to be after dark with a child is around Montmartre. I would skip the area altogether. There are much prettier churches and views that are as good or better in other places. A couple of good views are the top of the Arc de Triomphe (open air with a chest-high stone wall or the Tour Montparnasse, open air with high glass walls.