Congrats! I don't want to repeat everything I said earlier, but understand that Dickinson isn't necessarily going to match - their thinking is that Duquesne isn't as desirable as Dickinson, so they don't have to offer as much to be attractive. If Dickinson is her first choice and she would definitely go there if they bump her to the $20K scholarship, here's how I would frame her letter (not these exact words, but along these lines):
Dear Dickinson,
Thanks so much for the generous $15K merit scholarship. I am writing to ask if it's possible for you to re-evaluate my application in light of my ongoing academic success as a senior, and consider awarding me the $20K scholarship. Since my initial application, I have improved my GPA and continued to pursue a very demanding academic schedule. I have included an unofficial copy of my most recent transcript/report card, and would be happy to have my guidance counselor send you a formal version if you desire.
Dickinson is my first choice school. However, finances are a consideration for my family and I have been accepted with merit aid awards at other institutions that would make them much more affordable for me to attend. I don't expect Dickinson to match those offers. But if the College is able to award me the additional $5k a year merit award in light of my ongoing academic achievement, it would close the gap enough to enable me to enroll.
Thank you again, and please don't hesitate to contact me for any additional information.
Sincerely,..