。
She didn't notice that one of her buttons wasn't fastened properly.
Literal
She [topic-は] button's one [subject-が] firmly is-not-fastened [nominalizer-の] [to-に] did-not-notice.
This sentence stacks several patterns. ボタンの一つ ('one of the buttons') is the standard 'one of [a set]' construction. しっかり is a state mimetic for firmness or tightness — here modifying the fastening with 'properly, securely.' 留まっていない is the negative resultative: 'is not in a state of having been fastened.' The whole observation gets nominalized with の and marked with に as the content of the noticing verb 気がつく ('to notice'). Note 気がつく is intransitive in Japanese — what is noticed appears with に, not を. ボタン itself is a Portuguese loanword (from botão), an early borrowing from the Iberian-trader era of the 1500s — one of the oldest layers of European vocabulary in Japanese.