。
She sat quietly so as not to wake the baby.
Literal
She [topic-は] baby [object-を] wake-not [so as to-ように] quietly was-sitting.
起こさないように is the negative purpose pattern: 起こす (to wake someone up) → 起こさない (not wake) + ように (so as to) = 'so as not to wake.' 静かに (しずかに) is the adverbial form of 静か (quiet). The ~ていた form marks this as a sustained past action — she maintained her quiet sitting posture over a period of time.