Swimming pools in Paris

Our daughter in law and granddaughters used this indoor/outdoor piscine http://www.pailleron19.com/ in the 19e when we were staying in the neighborhood 2 years ago. They liked it and found it easy to use. You could combine this with an excursion to Canal St. Martin/Bassin de la Villette or Parc Buttes Charmont. Closer to your 5e neighborhood are these piscines Jean Taris or Josephine Baker.

Piscine Pontoise would be a good choice, since it's in the 5eme arrondissement.

A bonus is that it's open at night as well as during the day.
Remember that everyone needs to wear a swim cap, men have to wear the equivalent of Speedos, bring your own towel and toiletries and a 1 EU coin for the locker (refundable). Here's the address and hours of operation:
http://equipement.paris.fr/piscine-pontoise-2918

The weekly guide Time Out has this current feature:
https://www.timeout.com/paris/en/sport-and-fitness/swimming-pools-in-paris

The next site has general advice. Note its comments about men wearing caps and briefs not shorts. I've seen arguments about this on-line so maybe your hotel can check once you have picked out a location. http://www.travel-lists.co.uk/travel/articles/488-paris-swimming-pools-a-users-guide

Here is the official municipal site: https://meslieux.paris.fr/piscines
It's in French but has a map and is up-to-date on places that might be closed for maintenance.
One good-sized place marked for laps is Roger le Gall but three times a week it is used for nude swimming.

There is a swimming pool at the Butte aux Cailles, near Place d'italia (just 2-3 blocks from the metro) . It's in a beautiful, listed building and is very reasonable -

Address : 5, place Paul Verlaine 75013 Paris
Opening Times : 10 am to 8 pm every day except Sunday when it closes at 6 pm
Tel : +33 (0)1 45 89 60 05
Price : 3€ (adults) / 1.70€ (children)

You will need to bring a bathing cap and towel but they do have changing rooms and lockers.

My thanks for all the thoughtful information

Note that almost all municipal pools are used by school children during the day, and there are times when aquagym classes, etc are held, so public access is restricted to only a few lanes - or will be unavailable during these times.
You can swim laps in the "fast lanes", which is probably the best choice for you.
Avoid the "kiddie lanes" at all costs.

Here's an article from a Paris blogger who's a devotee of swimming with a kind of "how-to" about Paris pools:

http://www.pret-a-voyager.com/2012/02/paris-practique-french-swimming-pools/

She also participated in a book by Lisa Congdon about Paris swimming pools - she mentions here some Instagram hashtags that you can search to see photos of pools in Paris, and this blog post includes a blog shot of (some of?) the pages on Paris.

http://www.pret-a-voyager.com/2016/04/the-joy-of-swimming/

For your swimming pool research!

Kim, great blog for the uninitiated. When our extended family went, our son chose not to go because of the speedo requirement--that's just him...Some of the girls had to buy swim caps but they said they had inexpensive ones to sell at the counter or in a vending machine. My daughter in law remarked about all of the body cleansing showers and foot bathing basins prior to entering the actual pool. She said it was the most ritualistic locker room experience they'd ever experienced. However they went swimming at our London area house too and said it was similar but with out the speedo requirement for men--our son went along and wore his normal swimsuit.

Good advice about public pool requirements. It's not so much a speedo requirement as something that is only for swimming and can't be used for anything else - a speedo is the most common answer. Board shorts, conceivably, could be worn for other outside activities.

I believe all the public pools have vending machines in case you need a cap or swimsuit.

Private pools (e.g. hotel pools) don't have the same requirement, so they can allow whatever they choose.

Women can wear whatever they like.
Men have to keep things tightly under wraps, so to speak. I've seen men and boys turned away for wearing suits that were not form-fitting.