A couple of different kinds of restaurants are being discussed here.
Kiki's Bistro, La Sardine, Brasserie Jo, and Bistro Campagne are all
French bistros. French bistros in Chicago are almost all casual and moderately-priced (entrees typically no more than twenty-ish) restaurants. Their menus are all quite similar to each other, and typically include classic dishes like onion soup, escargots, duck breast, steak au poivre, coq au vin, sole meuniere, and for desserts, tartes, mousses, and profiteroles. You can see a fairly complete list of the French bistros in Chicagoland, with links to their websites,
here.
On the other hand, MK, Avenues, Charlie Trotter's, North Pond, and L2O are generally considered
contemporary American restaurants. Contemporary American is an inclusive term and many such places serve food that is not strictly American, but rather, has influences from cuisines around the world, including French cuisine, both classic (as you would find at a French bistro) and contemporary (as at places like Everest). Their menus tend to be individualized, reflecting the creativity of their chefs. They also vary in price, with Avenues, Trotter's, and L2O at the high end, while North Pond and MK are not quite as expensive, but still more than most French bistros. Some of our contemporary American restaurants, such as Cafe des Architectes or Sweets and Savories, have more French influence than others, but they are still, at their core, very different from the classic French bistros.
Nothing wrong with either category, or with any of these places; they're all very good indeed. Check out the menus on their websites to see which sound like what you're looking for.
If you're looking for the best value, some of the French bistros have specials on one night of the week, such as La Sardine's and Le Bouchon's Tuesday nights with any three courses for $25, and a similar special on Wednesdays at Bistro Bordeaux. A few of the contemporary American places also have specials, notably Cafe des Architectes with their $29 "neighborhood friends menu" Sundays through Tuesdays (their everyday prix fixe is $42, still not bad), and Sweets and Savories with their $29 prix fixe every night.
Since you're looking for French influenced food, you might also consider the restaurants which combine contemporary Mexican cuisine with French influence; Mexique, in West Town, and Sabor Saveur, in Wicker Park, fall into this category. Their menus combine elements of contemporary American cuisine with Mexican ingredients and French preparation techniques and sauces.