Who wrote "Taverns, exchanges, bridewells, drawing-rooms", and what is
the title of the popem?

The author of this work is the English poet Edward Young
(1683-1765)and the title of the poem is Satire V. On Women.  The
complete set of seven satires appeared under the title "The Universal
Passion".  It can be found in his Complete Works published in 1854.

The line quoted is line 25 of the poem.  

"The Satires were originally published separately in folio under the
title of The Universal Passion. These passages fix the appearance of
the first to about 1725, the time at which it came out. As Young
seldom suffered his pen to dry after he had once dipped it in poetry,
we may conclude that he began his Satires soon after he had written
the Paraphrase on Job. The last Satire was certainly finished in the
beginning of the year 1726…  The fifth Satire, On Women, was not
published till 1727."
http://www.hn.psu.edu/Faculty/KKemmerer/poets/young/default.html (from
Samuel Johnson's Life of Edward Young)

"Between 1725 and 1728 Young published a series of seven satires on
The Universal Passion. They were dedicated to the duke of Dorset, Bubb
Dodington (afterwards Lord Melcombe), Sir Spencer Compton, Lady
Elizabeth Germain and Sir Robert Walpole, and were collected in 1728
as Love of Fame, the Universal Passion. This is qualified by Samuel
Johnson as a " very great performance," and abounds in striking and
pithy couplets. Herbert Croft asserted that Young made £3000 by his
satires, which compensated losses he had suffered in the South Sea
Bubble." http://75.1911encyclopedia.org/Y/YO/YOUNG_EDWARD.htm (from
the Biography of Edward Young in the 1911 edition of the Encyclopedia
Britannica)

Searching on "bridewell" gave the following definition from a glossary
of words used by Charles Dickens: "bridewell - contraction of St.
Bridget's Well in London, site of a prison until 1869. Bridewell came
to mean any house of correction."
http://www.fidnet.com/~dap1955/dickens/glossary.html

Unfortunately, the poem is not available freely online and for that
reason I cannot supply you with the text.  However, the English Poetry
database of Chadwyck Healy, available through Proquest, contains the
full text of the Complete Works of Edward Young
(http://collections.chadwyck.com/html/ep2/bibliography/y.htm).
Membership of some libraries (public, school or college) will entitle
you to access this resource.

Search strategy: Once I had established the authorship by using
Proquest, I obtained further information by searching on 1. "Edward
Young" "On Women", 2. "Edward Young" "Universal Passion", 3. bridewell

Excellent, but gave me much more than I required!