I do not practice in MN, but your question is general enough that I may offer some assistance. I agree with the first answer in that, you should send a copy of your request to the prosecutor, as well as the judge. But, I disagree with the first answer on the need for a formal motion. Often times, courts are more lenient and understanding of persons representing themselves from jail and do not require all the formalities of a motion. Your probation officer's advice is likely correct, and you may write a letter to the judge.
In your letter, you should explain everything stated in your post. If you have any medical records or other documents to support your ex's medical issues, you should attach them to the motion. Perhaps, your ex would be willing to write a letter requesting your early release, as well, so you can help with the children again.
Further, you should advise your court of facts that paint you as a stable, secure person, with ties to the community. So, you age, education, where you live, how many children you have, employment history, etc. Judges also tend to care about whether you have any prior criminal history and whether you were cooperative with the police in this case.
For further, more specific advice, consult with a local, experienced criminal defense attorney. Your chances of being released early increase exponentially with an attorney on your side. But, the cost may outweigh the benefit in this case.
I don't practice in MN. Generally, you can't write directly to the judge. You should file a motion and serve it on the DA called Motion for Modification. Request change, give reasons, set it for a hearing date.
Edward J. Blum