At the end of the story, when Jane is crawling around the room and she "creeps" right over her husband who fainted, she says "I've got out at last." Does this mean that Jane has prevailed, or would you say that she has not been liberated (and has not prevailed) because now she really is insane? Not sure how to view this ending. She feels she has escaped, but her mental state has declined so greatly that she actually is mad now. Please explain if Jane's ending is a success regarding her inner struggle and independence or not!

We know that the narrator first loathed the wallpaper and then attributed consciousness and intention to it; then, she begins to ponder it; then, she actually comes to think it is helping her to feel better.  Near the end, she begins to feel that there is a woman trapped in the wallpaper, behind the bars that form the outermost layer.  The narrator says, "Through watching so much at night, when [the wallpaper] changes so, I have finally found out.  The front pattern does move—and no wonder!  The woman behind shakes it!" 
Keep in mind that, by the end of the story, the narrator's husband has confined her—essentially imprisoned her—in this house and this room.  She has not been allowed to see friends, to read or write, and so forth. Moreover, it is clear that this is a highly intelligent and creative woman.  There are bars on her windows, a gate at the top of the stairs, "rings and things" in the wall (perhaps to chain her and keep her still?), and the bed is nailed down.  Her husband's intentions may be good, but make no mistake: she is a prisoner.  She knows it.  She trusts him less and less, and by the end, she says, "He asked me all sorts of questions, too, and pretended to be very loving and kind.  As if I couldn't see through him!"
Throughout the story, the narrator is not always able to recognize herself as responsible for the ravages she sees in the room.  She talks about the "even smooch" around the walls, down by the mopboards, but she does not realize that she is the one who made it.  She talks about the bed being "fairly gnawed," blaming it on the children she thinks lived there before, only to tell us later that she "got so angry [she] bit off a little piece at one corner."  
When the narrator makes up her mind to help the woman in the wallpaper, it seems to empower her in a way that she is not able to help herself.  She says,

As soon as it was moonlight and that poor thing began to crawl and shake the pattern, I got up and ran to help her.  I pulled and she shook, I shook and she pulled, and before morning we had peeled off yards of that paper.

The narrator may not be able to free herself from her prison, but—in her mind—she can free this other woman.  Therefore, it is not so surprising when, toward the end of the story, she says, "I don't like to look out of the windows even—there are so many of those creeping women, and they creep so fast.  I wonder if they all come out of that wall-paper as I did?"  Throughout the story, the narrator, Jane, seems to have done things that she does not realize she did, paving the way for this total break in her identity. She begins to identify as the woman she believed she freed from the wallpaper.  This woman, from the wallpaper, is free; she was not.  She becomes the wallpaper woman in order to be free. 
On one hand, she believes herself to be free (which is good), but it is only because her mental health has degenerated to such a significant extent that she does not know who she really is (which is bad).  Jane achieves independence but only through a complete mental break.  Ultimately, I do not think Gilman wants us to be happy for Jane; a bright and vibrant woman has completely lost herself as a result of the backwards thinking and condescending treatment of her husband and doctors.