Он так хорош в письме. Note that "письмо"(writing) is noun while "писать"(to write) is a verb.
While you might hear this usage in colloquial speech or in some English dialects, it's not considered correct. "Well" is the adverbial form of the adjective "good."
Notable exceptions: food smells & tastes good, people look good, ideas sound good and lids should be kept good and tight.
Your answer is perfect, but DL does not seem to be programmed to tolerate such syntax transformations.
The word хорошо in the audio is pronounced as if it was a question. To make the sentence sound as an exclamation you should raise your pitch from он to так, then make it fall either all the way through the rest of the sentence or at least as far as the end of the word хорошо. In the latter case, you would say пи- in пишет at a very high pitch and pronounce -шет at a mid-low pitch.
There’s a difference between “writes very well” which is a dull trivial statement, and “writes so well!” which is an emotional exclamation.
In response to "type what you hear" I typed: он так хорошо пишет but it was marked wrong. What is wrong with this? Can anyone help? Duo says: Correct solution: Он так хорошо пишет!