It's done!
This morning I called the 'Studentensekretariat' which is in charge of students changing their addresses, changing the majors etc. between courses - I had Latin 1 and for French 'Jewish characters in literature and film'... I explained that I had received this letter about me being accepted for law, but that it is at my parents and that I had no chance to get the letter and show thzem in time. I was told that a simple letter written by me explaining the situation would be enough, but they also reminded me not to forget my student ID.
So I went to the 'Studentensekretariat' in the afternoon and got my subject changed and now I'm officially a student of law at the Goethe-University Frankfurt/Main!
It's so incredible and I'm soooo happy!!!