。
I wrote down her address so I wouldn't forget.
Literal
Forget-not [so-that-ように] her address [object-を] noted-down.
The ~ないように + 書き留める frame ('write down so as not to [verb]') is the standard Japanese way to express purpose for note-taking. 書き留める is a compound verb (write + retain) — a common compound-verb pattern with 留める attaching to action verbs to mean 'fix, retain, hold in place.' 彼女 here means 'she/her' as a third-person pronoun, not 'girlfriend' — context decides which sense applies. Subject is omitted; context supplies it.