Thirty-third and State, that's right by Crown Hall—one of the most architecturally significant structures of the 20th century. Here are some suggestions within walking distance of Mies's masterpiece.
Ferro's (200 W 31st) has been serving decent Italian beef and lemonade for ages. I've been meaning to get back there to see how much of the Wiviott influence rubbed off.
Mr Spanky's (335 W 31st) is well worth a visit for sandwiches featuring their excellent housemade bacon or the daily special. I need to write something about this fine little place.
A little farther along 31st you can get a Polish or a chop at Maxwell Street Depot (411 W 31st).
There's more along 31st (Scoops Ice Cream, Pancho Pistolas, Freddie's, Northern City, even Pleasant House) but they're more than 15 minutes away.
Or you could head south to 35th. If you walk fast you could get to 35th Street Red Hots (500 W 35th) for some of the city's better minimally dressed dogs and fresh-cut fries.
The place I'm most interested in is Mama Lou's Comfort Kitchen (124 E 35th), a relatively new soul food place. I stopped in one day but the smothered chicken wasn't ready yet.
Heading east on 35th you'll find a Harold's Chicken Shack (321 E 35th). I haven't been to #19 yet but I approve of the artwork
Southtown Sub (240 E 35th) has an extensive menu of classic Chicago bad-for-you foods.
On King, just south of 35th is Mississippi Rick's (3511 S King). I've only been to the old one (now closed) but this place has its fans.
A little farther south is Pearl's Place (3901 S Michigan) for fried chicken and peach cobbler. I understand they've reinstated their lunch buffet.
Pearl's might be a little far (Google tells me it's a 19 minute walk) but you could take the Green Line one stop to Indiana. Also, Chicago's Home of Chicken & Waffles (3947 S King; I haven't tried it) is only a couple blocks from that L stop.
If you're willing to ride 15 minutes on the L, that opens up a big chunk of the city to you. I won't make a list for you but within your 15 minute limit you can choose from Daley's at one end (Cottage Grove) or Publican Quality Meats at the other (Morgan).