After theater suggestions
For dessert and coffee, I'd recommend the Bar Room at the Modern. They have a decent cocktail list. It's not super quiet, though. Maybe a (very) high end restaurant's bar like Le Bernardin or Jean George would be quiet enough?
What day of the week is this? When are you coming?
For cocktails, there has been a recent resurgence of downtown mixologists doing hotel and other midtown bars. However, it's often difficult to find a quiet bar anywhere in Manhattan especially on a Thursday, Friday, or Saturday.
44 at the Royalton Hotel with a list done by famous bartenders from all over -- Richard Boccato (Dutch Kills, Painkiller), John Lermayer (Florida Room, South Beach; Woodward, Boston), Simon Ford, Willy Shine, Misty Kalkofen (Drink, Boston), and Eric Alperin (Varnish, LA)
Lamb's Club has drinks by Sasha Petraske but they are much pricier than comparable places nearby
Lantern's Keep in the Iroquois Hotel by Meghan from Raines Law Room
Rum House by the Ward III folks though it most likely won't be quiet at all, and they usually have live jazz
Bar Downstairs (cellar bar) at the Andaz by Toby Maloney of the Violet Hour in Chicago
1 Central Park W, New York, NY 10023
155 W. 51st St., New York, NY 10019
9 West 53rd Street, New York, NY 10019
Raines Law Room
48 W 17th St, New York, NY 10011
111 Reade St, New York, NY 10013
228 W 47th St, New York, NY 10036
The Lambs Club
132 W 44th St, New York, NY 10036
44 W 44th St, New York, NY 10036
49 W 44th St, New York, NY 10036
The Bar Downstairs
485 5th Ave, New York, NY 10018