Why do you need to write a letter to your mother? You don't say what your punishment is, but in all actuality if you forgot an important doctor's appointment and she now needs to reschedule it (not always easy), she's upset with you. If you need a medication refill, that is essential and she's probably hoping that you learn how important it is to keep track of your appointments.
Is this an apology letter? Is this part of your punishment? Are you a parent and that's why you are on this site? If you aren't a parent, this isn't really an appropriate site for you to join.
If it's an apology letter, try something like:
Dear Mom,
I am sorry for missing my doctor's appointment yesterday. I had the dates confused and was really excited about my interview. I missed my interview because I didn't ask the right questions on the bus and know now that if I am unsure, I should ask.
Love,
_______________________
Really, though- a conversation about what happened is more appropriate in this situation.