The woman writes one single letter - seule lettre, that's what i got here. Otherwise ne que is used as the english "but one" or "but a single" in "She wrote but a single letter" in which the word "nothing" is excluded
Basically saying she is writing one lonely letter (a single letter). If you followed 'lettre' with 'seule' it would mean she's writing the letter by herself, alone. Can you say 'la femme ecrit une seule lettre seule?
That would mean "The woman writes a lonely letter." In this case, 'one letter' is quantitative and not qualitative. Being lonely is a quality, which is totally different than the quantity of 'one letter'. There could be many lonely letters!