忘れてはいけないので彼女の宛名を書き留めた。

。

I wrote down her address so I wouldn't forget.

Literal

Forget-[must-not-てはいけない] [because-ので], her [possessive-の] address [object-を] jotted-down

Note the causal chain: 忘れてはいけない ('mustn't forget') + ので ('because') gives 'because (I) mustn't forget...'. 宛名 specifically means the name written as an addressee (as on a letter or package), not just 'address'. 書き留める (write-stop) = to jot down so it stays.