Based on my understanding of your question in the image you linked to, what you want is to obtain a handle to the edit control. If that is so, it's very easy...
I have a handle on the window of the browser, how can I find a child window within it (given that I have the text of the child window)? I've tried EnumChildWindow(), but that only enumerates the child windows of the parent windows, and doesn't deal with child windows of child windows.