Mick's inner room is a metaphor that Mick uses to describe, well, herself. Or her true, inner self at any rate. Mick has basically partitioned off her mind into two sections – there's her outer self, which is concerned with school and family and money. Physical, real-world things. And then there's her inner self, which is filled with music, plans, dreams of far off places, and of course, Mr. Singer.
Mick's actually practicing some pretty deep philosophy without even realizing it. And her inner room represents a powerful theme in the book: alienation. Every character in the book has their own "inside room" which isolates them from the rest of the characters. And, of course, everyone gives Singer an all-access pass to their "inside room," not that he knows how to cash it in.
A lot of modernist literature focuses on how other people define who you are, and how it's almost impossible for anyone else to really know you. (Kind of a downer, but hey, don't shoot the messenger.) Mick is able to protect her inner thoughts in her mental inside room, but she also cuts herself off from other people by internalizing who she actually is. Inner rooms symbolize the barriers that all of our characters put up – intentionally or not – to keep others out.