I'm in the same situation. I am rejected from all grad schools I applied, and I plan to reapply at the same school. Actually I emailed to the program director who interviewed me (and rejected later) to ask for my application flaw and suggestion, but he did not reply. I'm not sure if the process of admission decision must be kept in secret. Now I have no idea which parts of my application should be improved, except GRE score. For letter of recommendations, I've already informed my rejection to my recommenders and ask them for writing next round, and they're willing to help. So I think in your case, your prof. won't mind to write the letter again.
BTW, I also need advice from others about reapplying the same school (school X). In my situation, I received the interview invitation from school X, and later I was rejected. So I don't know if I should apply since I don't know whether the problem is that my GRE is poor or they don't like my research (which I cannot fix it as JungWild&Free said). Thank you for all comments.