How should I reply to this Professor's email?
Can anyone suggest a draft of the email that I can send to this Professor? 


Reply to the Professor's original email, so that your response appears in the same thread; perhaps include the original snippet, i.e., 


  Thanks for writing and for the great ideas -- I see a lot of energy and promise in your thoughts, and I encourage you to try to pursue some of these questions -- most, in fact, are exciting and open research areas. I also encourage you to apply for PhD at YYY University.


and write something along the lines of: 

Further to your positive encouragement, I have submitted my PhD application to YYY University.

Thereafter, it depends on your agenda. E.g., you might want to reopen discussion of  joining her group, for which you could use:

Given the seeming alignment between our research interests, I would like to join your group. Can we discuss that possibility, perhaps by telephone?