Yup, hopefully, your definition of "within a short distance" would include The Brown Sack. If it's not like "peak" lunch time, you'll do pretty good there with the little one.
Also a coupla blocks up Kedzie is Lula Cafe.
Brown Sack is a tiny lil sammich shop with (IMO) fantastic food. Soups, sides, sammiches, shakes - it's all good.
Lula is a lot more restauranty, but still has a neighborhood "place" kinda vibe.
I'd be happy to go to either.
We cannot be friends if you do not know the difference between Mayo and Miracle Whip.