Near Hotel Felix?
I'd second Xoco as well - it's just a few blocks from the Felix and is quite wonderful ... my coworkers eat there almost everyday for breakfast and we all go in the afternoon for a snack. It's fun and lively ... great spot.
We put up business guests at the Felix on a regular basis and really like the hotel - nice mix of trendy with friendly service - and they have a good restaurant on site.
You might also like to check the restaurant menu at Sable which is just over on State St - the new home of one of our early New American chefs, Heather Terhune, they have a $15 lunch special that has us going back often ... and the inside booths face the open kitchen which can be quite entertaining.
One more - rather bar sceney at night but we also often lunch at Mercadito which does a really nice selection of tacos, salsas and ceviches ... just around the corner from all the great Bayless places but a very different take on Mexican and reasonably priced. (When I take my team, we order a sampler of the salsas and of the ceviches and then each get a taco selection - cost below $20 per person and very good food.
Oh, and if you are hitting Nordstroms for shopping, their cafe on the fourth floor has some very tasty food if you don't want to wander too far from the shopping.
108 W. Kinzie, Chicago, IL 60610
For pizza... From the hotel, the original Uno and Due ( www.unos.com ) are three blocks east and three blocks south. Lou Malnati's ( www.loumalnatis.com ) is one block west and five blocks south. Pizano's ( www.pizanoschicago.com ) is two blocks east and three blocks north. All are excellent (and were all established by members of the same family). Phone ahead with your pizza order to avoid waiting 30-45 minutes while seated for your pizza to bake.
Other good but not terribly expensive places near the hotel include:
Orange (breakfast - N. Clark St. location) - www.orangerestaurantchicago.com
Portillo's (Chicago hot dogs and Italian beef sandwiches, Ontario St. location) - www.portillos.com
Cafe Iberico (tapas) - www.cafeiberico.com
Graham Elliot (contemporary American) - www.grahamelliot.com
Roy's (Hawaiian) - www.roysrestaurant.com
Carson's (barbecue, steaks) - www.ribs.com
Quartino (Italian small plates) - www.quartinochicago.com
Pops for Champagne (wine bar) - www.popsforchampagne.com
Nacional 27 (Latin fusion) - www.n27chicago.com
Sunda (pan-Asian) - www.sundachicago.com
Naha (contemporary American) - www.naha-chicago.com
All of the above are within a five minute walk of the hotel, and most of them accept reservations on Opentable.com
If you're willing to walk a few blocks- I'd also suggest Xoco, Rick Bayless' order at the counter place with Mexican sandwiches etc and La Madia for delicious thin crust pizza and Italian. Also- you might want to clarify what your definition of expensive is or make sure you check out the menus beforehand- of all of the really good suggestions provided above- I would consider Naha somewhat expensive, with entrees in the mid 30s and up.