To make passive sentences first you've got to identify your subject and your object. The subject is the person or thing do the activity. In my lessons I usually use coloured markers, I wish I could use colours on the discussion boards!
In your first sentence your subject is "He"(him) and your object is letters.
You need to start your sentence with the object:
Then you need to replace the main verb with be and copy the tense exactly
has been writing = has been being
Then add the 3rd form of the main verb afterwards
has been being written
So lets put our sentence together!
The letters has been being written.
Oh no! There's a mistake! "He" is singular but "letters" is plural so we need to change has to have
The letters have been being written.
At the end if you want to include the subject you can add "by him" but usually the subject is not very important - that's why we changed it into the passive voice! Can you do the second one on your own using this guide? I'll correct it if it's not right.
Let me know if you have a question about this, I look forward to seeing your second sentence rewritten!