Is it to be furnished by John Merrick's nieces, I wonder ?" Willing Square is a new district, crowded with fashionable apartment houses.

That is, they are called fashionable by their builders and owners and accepted as such by their would-be fashionable occupants.

Diana knew at least two good families resident in Willing Square, and though she smiled grimly at the rows of "oppressively new and vulgar" buildings, she still was not ashamed to have her equipage seen waiting there.
Number 3708 Willing Square is a very substantial and cozy appearing apartment building owned in fee by Miss Patricia Doyle.

Diana was unaware of this fact, but rang the Doyle bell and ascended to the second floor.
A maid received her with the announcement that Miss Doyle had "just stepped out," but was somewhere in the building.

Would the visitor care to wait a few minutes?
Yes; Diana decided she would wait.