If I have to use 'a flight of ' as quantifier to desribe stairs

It seems to me that a "flight of steps" was once a fresh image, an imaginative conception of someone's passage up a stairway as flying by means of walking: a flight of steps. This wording used to be more common than "flight of stairs," but perhaps as the expression became familiar and lost all force as an image we stopped preferring the first expression over the second.

(Of course this could be wrong; the word 'flight' is also used for other sorts of groups, and may have meant simply "a group of steps.")

Anyhow, I would say that you do not have to use 'a flight of' to describe stairs; you could say (with little variations in meaning) a staircase or a stairway instead. And it seems to me the most natural way to include the landing in the description is explicitly: a staircase (or a stairway) with a landing. (I did find some uses of your expressions "double staircase" or "double stairway," but the "flight of" expressions were around 100x as common.)