Help with what to do in Madrid

lots of info here: https://www.ricksteves.com/europe/spain/madrid

Thanks, checked there. Similar to what I am already doing. Was looking for some additional ideas of how to spend time, especially if I am doing those things by myself.
Thanks

From Puerta del Sol you are close to the Plaza Mayor and the San Miguel market. Are you going to the Royal Palace and the Almudena cathedral across the plaza from the palace? That is also not too very far from your area. How about El Retiro Park? Lots to see there. If you are flea market fans, try El Rastro, held on Sundays, but expect crowds.

Will you be on a tour to actually visit sites or is it a bus tour? With the list of streets you give it makes me wonder if it is just a driving tour around the city. Do you like art? I see you plan to visit El Prado but the Reina Sofia museum is also nearby. If you are big soccer fans, try a visit to Santiago Bernebeu stadium, home of Real Madrid football club.

I see your itinerary includes Segovia (which you will enjoy very much) but are you going to any other destinations in Spain after Madrid? If so, save the flamenco show for Andalucía.

I like to recommend nearby Toledo but I don't think you have enough time. Same for El Escorial. With so little free time i would stay in the city.

Depends on what you like to do. The Reina Sofia museum has free evening hours. Retiro Park is nice for walking and people watching. Near Cibeles is the Fuente de Neptuno (a fountain similar to Cibeles), and the Lucky Frog statue Both are photo ops. Plaza Mayor, Plaza de Espana, Royal palace, temple of Debod, Ermita de San Antonio de la Florida with frescos by Francisco de Goya, Puente (bridge) and Puerta (gate) de Toledo. (Similar to Alcala gate). By the royal palace is the cathedral de Santa Maria la real de la Almudena. It’s ok for a short visit. A couple minutes walk from Puerta del Sol in the direction of the direction of the Plaza Mayor, and just off Calle Mayor is Chocolateria San Gines. Good churros and chocolate.

For that free morning, you could check out the Palacio Real de Madrid. It's the official residence of the Spanish Monarchy, although they don't live there anymore. It is one of the largest royal palaces in the world. The interior has many rooms that are very richly decorated, each with their own theme. The Palace also houses the Royal Armoury, and consists of the various arms and armour of the Spanish Kings from the 12th century to the 20th.

I've never been to a Flamenco show myself, but it seems like a nice option for the free afternoon you have. I do know that the general rule is that the best Flamenco shows are the ones without the dinner, just the show.

thanks for the ideas.
I have read that Retro park was not safe to walk around in. Is this true?
Also, I do not think we go into the Palace. Is it worth some time?
Only in Madrid, no other location in Spain, so not sure the Flamenco show/dinner is worth it. Food at that event, good?

I'm not sure if the places you list are those parts of a tour you've already arranged or what you're considering to do on your own to fill the spare morning and afternoon. If the latter, then I'm unsure if Segovia is practical given the need to get to/from railway stations.

Other sights you might consider are Mercado de San Miguel, the royal basilica (better than the cathedral), the royal palace (huge but dull, there are so many better palaces in Spain, let alone Britain & Europe), and the observation tower Faro de Moncloa.

As for Flamenco, it's just another faintly ridiculous folk dance. Do you like Schuhplattler or Morris or American line dancing? If so, you might like Flamenco too. Personally, I'd do a tapa tour to enjoy the food but skip the palmas.

PS Templo de Debod mentioned above by Jaimeelsabio is great, but recently it was closed and the pools drained, so might be worth checking it is back up to snuff before going.

Here're some other museum suggestions.

The Archeology Museum was my favorite of all the museums we visited. It is very informative on the human history of the Iberian peninsula from prehistoric to the Christian era. It is free on Sunday morning. Rent the multimedia guide and chose the Essential 33 Route. Takes a couple of hours to see this section.
Another is the Sorolla Museum. The garden is beautiful. The audioguide has the same information as the signs in the rooms but
it is easier to listen while looking at the paintings. It will take only about 60 mins (90 max) to visit.
The Royal Palace is worth it if you are into royal residences. It is the 3rd great palace after Versailles and Schoenbrunn.
IMHO, I don't think flamenco show and dinner is worth it in Madrid. Better to see a show in Sevilla. Skip the dinner.
We walked in Retiro Park on Sunday afternoon. We did not feel threatened at all. There were families with small children playing. A nicer, smaller park is the nearby Botanical Garden. For limited time in town, I would chose the latter if you want to walk in a park.
Be sure to try Valor hot chocolate and churros. There are several locations. Check on their website for the one nearest you.
Or go to San Gines which is open 24/7.

We loved the Reina Sofia Museum! Even though I am not a huge fan of modern art, I thought it was fascinating. And Picasso's Guernica is a must see! We felt very safe spending several hours walking around Retiro Park and drinking Sangria outside by the lake. In January I assume it will be too cold to sit outside, but it is a beautiful park, although not sure if flowers will be blooming. The Crystal Palace in the park is gorgeous! We also enjoyed the Royal Palace, and thought the inside was beautiful, interesting and definitely worth it. We rented audioguides.

We did not have time for the Thyssen museum, but I've heard good things about it. The top 3 art museums in Madrid are the Prado, Reina Sofia, and Thyssen.

The Thyssen Museum is "just up the street" from the Prado. It depends on what periods of painting most interest you. The Prado more or less stops in mid-19th century, where the Thyssen begins, with lots of Impressionists, then the Reina Sofia covers the periods after that. The excellent Naval Museum is "just up the street" from the Thyssen and well worth a couple of hours (bring your passport). The REina Sofia is another short walk from the Prado, in the other direction. All are walking distance from Puerta del Sol.

Flamenco is an Andalucian "thing" and I've only been to performances in Sevilla. The best ones are those that are strictly performance - no drinks, no food. I don't know if there are such shows in Madrid, if there are, it could be worthwhile. I would avoid a flamenco show that included dinner, as too touristy.

I've visited Spain mostly in February and found the gardens most underwhelming, most looked like they'd been pruned back with great gusto. I loved grazing through the San Miguel market.

Agree with Carlos. The Royal Palace in Madrid is spectacular. And it's not so big as to be unmanageable, unlike Versailles. It would be my #2 sight in Madrid, after the Prado.