While I can't find any specific rules on the difference, I can tell you what the different words imply to a native speaker.
When you say 'Hon bor mitt i staden' it feels more general, as if she lives just anywhere not in the outskirts of the city. If someone, for example, asked you if she could get to the subway easily, you could reply with 'Ja, hon bor mitt i staden'. This does not necessarily mean that she lives by the exact center of the city.
'Hon bor i mitten av staden' (remember the 'i') is more like a specific location, as in the actual middle of the city. If you were to look at a map of Stockholm you could say that Riksdagshuset is 'i mitten av staden', while someone who lives by Sankt Eriksplan could be said to live 'mitt i staden'.
Hope this helps!