First of all, the proper spelling in this situation is "principal."
If you know his/her name, include it in the salutation (Dear Mr. _______). If you don't know the person, Dear Sir or Madam is correct. (But if there is any way to find out the person's name, that always makes a better impression.)
Make the letter cordial, but brief and to the point. Introduce yourself and your child in the first paragraph. "My name is Kini Bunkport (or whatever), and I would like to enroll my daughter, Kimmy, in ____________ (insert the name of the school).
In the second paragraph, explain why you want your child to attend that school, and mention any special qualities your child has that makes her a suitable student. "I'm well aware of ___________ School's fine academic reputation, and believe that Kimmy would thrive in that environment. She excels in Mathematics and Science, and is an attentive and conscientious student. I have transcripts from her previous school, and letters of recommendation from some of her former teachers that I would be pleased to submit for your perusal.
I realize that the deadline for applications has passed, but I believe that if we could discuss this matter in person, you'd agree that Kimmy should be given consideration. I look forward to hearing from you at your earliest convenience.
Obviously, this is a very general outline of a generic letter. You'd fill in your own child's name, etc, and add any extra details that pertain to the particular school and your situation. I hope this example will give you a starting point, though. Good luck!