Translation:She is the professor I wrote to.
"She is the professor whom I have written" has not been accepted. Instead Duo suggested "She is the professor I wrote to." Now you can write to me, but you can also just write me, at least in American English:
https://www.oxfordlearnersdictionaries.com/definition/english/write?q=write (look for "write somebody" under the third meaning).
This needs to be added.
In German you can write a book, a letter, a sentence, but not a person. It would have to be "to the professor", which is expressed with the dativ case, instead of the accusative, which is only for direct objects like a letter etc.
See also http://german.speak7.com/german_articles.htm - "der" is not always masculine, it appears in multiple positions in the table.
"She is the professor to whom I wrote." This gets my vote.
"She is the professor that I wrote to." This seems ok too, but more awkward.
The "to" is required in Australian English and probably British English.
It appears that American English has allowed the "to" to be omitted.
To my ears "She is the professor that I wrote." sounds very wrong, but I think it probably sounds normal to an American.