Apartment owners have agents largely because they don't want the hassle of dealing directly with the public. Plus, they don't have the resources to find renters themselves as the agent does.
I couldn't find an actual website for the "Zeffirelli Apartments", but did find it to be on calle de Torija, 6 in Madrid. This is a very good area, mostly residential, but also very very close to the old center. Actually, it's part of the old center, but I mean it's away from the tourist center. It's close to the Senate building, Café de Chinitas flamenco tablao (my vote!), to OPERA and the Royal Palace. Good location, indeed, and in a nice, old building. You'll love it.
In some photos I see that it does seem to have an "outdoor" terrace/patio with a table and chairs, but I'm guessing it's part of the interior patio, essentially the air-shaft on/near the ground floor so don't expect much sunlight.
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