Madrid lodgings- anywhere else we should consider?

You’d need to check if it has an elevator since I don’t remember, but right by the Reina Sofia museum is the Sleep’N Atocha. It was a basic hotel but adequate if on a budget.

Edited: Our last trip to Madrid was in May 2017 and we paid €109 per night at Hotel Europa. It may be cheaper in April, so might fit the OP’s budget. We enjoy that hotel and location, especially after the renovations it did between 2015 and 2017.

Look at Hotel Europa on Puerta del Sol. Loved it. Modern and new. Inexpensive. Very soundproof rooms so you get the fun of Puerta del Sol but still can sleep!

On my third trip to Spain, after having spent the first two in Madrid at Hotel Europa, I planned to stay at Hostal Acapulco, which was recommended to me by another forum member (and friend). It was significantly cheaper than the Europa and I wanted to save the money. In the end, I decided not to visit Madrid on that trip.

Thank you. We are also considering Hostal Gonzalo. They do have a small elevator. Any thoughts on them?

We stayed in Apartamentos Juan Bravo in the Salamanca neighborhood. I've just checked on Bookings.com. The apartment is about $80/night in April, 2019. Last year I booked very early and got it for $65/night. It has elevators and also a laundry room. It was more convenient for us to have breakfast and dinner in the apartment, especially dinner which is so late in Spain. There are supermarkets, restaurants and bakeries around the corner on C/ del General Díaz Porlier. You can easily take the Metro across the street to get to all the major tourist sites. The area is where ordinary people live, not loaded with tourists. The restaurants and shops were not as crowded as the ones in the touristy areas, the prices were slightly cheaper and the staff friendlier. It's a quiet, safe neighborhood with tree-lines streets and public benches.
A classic dish to have when in Madrid is cocido Madrileño. Within walking distance is Casa Carola on C/ de Padilla, 54 which specializes in this dish. It's the only dish they serve. There is English instructions. The staff speaks English and very friendly. It's all you can eat and the food was delicious. All the other guests in the restaurant were speaking Spanish. For those readers who might be interested in this restaurant, the closest Metro stops are Diego de León or Lista.

I'm the one who recommended the Hostal Acapulco to Chani. The only potential problem is that, while my reservations were handled by email in fluent English without difficulty, many of the staff spoke little or no English. I can manage the basics in Spanish, but if you need regular access to an English-speaking desk staff for assistance, you may want to stay elsewhere.

Otherwise, I see no need to consider other lodgings, if you are looking for a basic and small, but clean and pleasant accommodation in a great location.

Thank you all. We do speak a little Spanish. I am leaning towards Hostal Acapulco. Seems like for our budget, we can’t do better.

We ended up staying at the Hostal Acapulco for 4 nights. It was a perfect place to stay and we would stay there again. We booked a twin room, not because we wanted a twin, bur it was a good size with a balcony and nice view.

Thanks for the feedback!