A month later isn't too bad, actually.
And I wouldn't go on and on about being crazy busy--excuses just make you look bad. And will remind her that she was pretty busy at the time that she helped you. Not a good vibe.
Simply say, "I'm sorry it has taken me so long."
And one advantage you *do* have is that some time has passed, and you can hopefully come up w/ more specific ways to tell her that her assistance was so helpful.
Tell her the conclusions your paper drew, or what you remember most from the time w/ her, or what grade or reaction you got to your work that was based on her input. Or even that you've had time to reflect and drawn some conclusions, learned some lessons, from her.
People are curious; they want to feel that they made an impact. Even if it's just to say, "I still tell people about that joke you made at lunch. I get a laugh every time."
Especially when it's late, I think people deserve a longer and more detailed thank-you letter. A letter, not a note. It really won't take all that much longer to write.