The previous poster was right, we can't steer you, but we can inform you about neighborhoods that have the most outdoor access. Anywhere by the Lake would let your daughter enjoy the best outdoor possibility in Chicago. And if she could be by the lake and large swaths of park than she would find the ultimate in city outdoor opportunity.
The lakeshore is full of high and mid rise rental buildings. Primarily, in regards to ease of access (less than 5 min walk) to the lake, park, and transport, there are buildings in Hyde Park and Oakwood on the south side (unfortunately other neighborhoods on the south side, closer to her work don't offer that combo of park, lake, highrise rental), then the South Loop all the way up to Rogers Park, within one block of the water. And within that large expanse are a variety of neighborhood feels that only your daughter will know which is right by taking a few different weekends and a bicycle/foot to explore.
Don't worry too much. Tens of thousands of young professionals move to Chicago every single year. Most from less populated regions. And they all do fine. Or they leave the city. I'm sure she'll be great. Come visit her.