I agree that the judge doesn't know who you are, except that you're someone before him who has admitted guilt to a crime. The probation report he will see is not going to be very much help in filling out that picture.
Letters to the judge from the defendant do not work in my experience. We cannot write one for you here, as that would be representation, which the site does not allow us to engage in. Besides, none of us knows you and if you're going to write a letter it should come from your heart. However, I can offer you some guidance about what to say and would be happy to proof and edit for you when it's done.
That said, you are represented by counsel and you should give nothing to the judge at all that hasn't first been looked over by your lawyer.
What I have found works better than a letter from a defendant i something called a pre-sentencing memorandum, which is prepared professionally by your lawyer and his staff. That's a detailed report showing your personal history and background, discussing that in light of your case and your possible sentence and asking for leniency in the sentence.
That's basically what you have already said you want to write. The difference is that this is prepared by professionals, that it will include and showcase all of the positive things you have done in your life, along with your present remorse about what got you in trouble and would include letters of character references from people who care about you.
I suggest talking to your lawyer about his preparing submitting a private preesentence report and tell him you can get references as to your character from people who care for you and do not want you to go to jail. If the lawyer won't do that, then write your letter, but attach character references as well. Make sure the judge does not get this on the day of your sentence because by then he will have already have his mind made up.