I’m going to echo what some are saying and deal with this person-to-person rather than writing a letter for two reasons. First, you do need to let the school know that you are having difficulties with the language and that it is coming across in writing. Second, you need to ask about how you can get an English tutor to help you with future papers. Finally, I think you need to be in a situation where there is an actual back-and-forth conversation. I think it will help.
My impression is that you knew you were being academically dishonest, but you weren’t exactly sure how to stop it. That needs to be addressed or you will end up in a similar situation again.
In colleges, US kids with learning issues can ask for accommodations. I’m not sure if these are granted for international students in such situations, but it might not hurt to ask for extra time to complete longer papers in light of both the medical situation and the language barrier. Writing a term paper can be difficult in your native language, trying to translate your thoughts into another language just adds to the difficulty - regardless of how fluent you are.
Oh - and take someone with you for support and to help translate (I really want to say articulate, but that’s a big English word) your thoughts. Don’t try to do this on your own.