Still, how do you get "why" from だろう? That isn't the only question word in English. What Japanese question word do you see in the sentence?
I think you may have read an explanation of いったい somewhere and maybe let the explanation color your interpretation more than necessary.
How would you interpret the sentence without いったい?
わたし は 、 北本 くん に どう 思わ れ て いる ん だろ う ?
Pay close attention to the grammar.
Edit:
Come to think of it.... try it without the だろう
わたし は 、 北本 くん に どう 思わ れ て いる ?