"to" isn't needed. "me" is an indirect object ... you bring a cake (direct object) [TO] me (indirect).
Or, "you brought me a cake for my birthday!!"
I think (I hate to say always in English ... it's never right ) that you can always rewrite a phrase with an indirect object (object will be a person), to use a preposition instead, typically "to" -- so either could work.
Thank you, Ebb, for your prompt reply. Somehow the end of it seemed unfinished to me Am I to understand the last phrase as "Bring to me that book?" or else "Bring that book to me"? Or maybe both?
P.S. I wonder why instead of changing the fonts colour some unwanted square bracketed captions pop up?