$20 per person in the Loop is a tough order, I think. But you might check out Pizano's -- they have a room downstairs and pizzas are convivial. Something a little more "girly" might be the Toni Patisserie -- but i doubt they have a private space. Another option might be the Sopraffina that's on Dearborn near Madison. My company has used it for events before -- and it was apparently a good call. I have no idea what price they'd charge -- but you could check it out.
Pizano's on Madison
61 E Madison St, Chicago, IL 60603
(312) 236-1777
Toni Patisserie
65 E Washington St, Chicago, IL 60602
(312) 726-2020
Sopraffina
10 N Dearborn St, Chicago, IL 60602
(312) 984-0044