This question really depends on what you mean by 'independence'.
Literal independence comes for Jane when she is able to escape her step mother by applying to be a nanny. She is able to leave the house she has been tormented in.
A deeper sense of independence may come for Jane later in the book. It is arguable whether Jane is truly independent, or bound by Rochester. You need to analyze the decisions she makes (her reaction to the lady in the attic, her return to the burning house) to determine if she is truly independent.
The novel is considered a "bildungsroman," or, loosely translated, a"coming of age story." Jane becomes 'independent' by the end of the novel because the circumstances of her life teacher her a number of powerful lessons:
1. Her true worth is intrinisic. She does not possess any of the qualities that were valued in Victorian ladies: She is dark, plain, and introspective. Yet she is also extremely intelligent, and she addresses the world around her with forthrightness.
2. She comes to know herself, and she demands to be taken for who she is--no more, no less. In fact, she is reunited with Rochester only after circumstances have made him dependent upon her.
3. In a very literal (Victorian) way, she becomes independent when she discovers that she is not a penniless orphan, and she comes into an inheritance that means she does not need Rochester or any other (wealthy) man to be her husband. It seems like a bit of a cop out. After all, Jane spends her life coming to personal power through very difficult lessons, and then she inherits money that would have solved her problems even if she were insipid. I suppose, though, that we must remember that this is the Victorian era, and no amount of intelligence or strength of character will insure a woman an income