How can I explain my situation to the underwriter in the form of a letter showing the debt in my name is paid by my daughter?

I think you are doing exactly what you should be doing. If you can show that for the last 2 years your daughter has been paying on the account, the underwriter should either 1) back out the car payment and not factor it in to the overall analysis or 2) consider as income the regular payment your daughter is giving you which should then be a wash with the car payment you are making on her behalf.

LEGAL DISCLAIMER
Mr. Larkin is licensed to practice law in CA. His response here does not constitute legal advice and does not create an attorney/ client relationship. The response is in the form of legal education and is intended to provide general information about the matter in question. Many times the questioner may leave out details which would make the reply unsuitable. Mr. Larkin strongly advises the questioner to confer with an attorney in their own state to acquire more information about the specifics of their case.