Sending a thank you e-mail to a professor as an external intern
I think the proper answer is "Just Do It." She will be delighted to hear from you. Tell her about your current success. Open and close with "thanks". That's all it takes really. 

Of course the things you say here would be excellent to include in your thank you. 

However, sometimes a more formal letter is called for, sent by ordinary mail. That is up to you, of course.