。
She added in the letter that she would write again soon.
Literal
She [topic-は] letter [to-に] soon again letter will-write [quotative-と] added.
付け加える (つけくわえる) is a compound verb meaning 'to add' — 付ける (attach) + 加える (add). お便りする (to write/correspond) is a polite, humble way to say 'to write a letter' — the お prefix and する make it a respectful expression for one's own letter-writing. 近く (soon/shortly) here means 'in the near future.' The quoted content (近くまたお便りします) is polite speech embedded within the narrative.