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She didn't notice the cold until she opened the door.
Literal
She [topic-は] door [object-を] open [until-まで] coldness [to-に] [hadn't been-noticing-気付いてなかった].
Two key pieces. 気付く ('notice, become aware') takes its noticed thing with に, not を — the noticed thing is the target of attention rather than a manipulated object. The form 気付いてなかった is the casual contraction of 気付いていなかった: in spoken/casual Japanese, the い of ~ている often elides, giving ~てる, ~てた, etc. Putting it together: 'until-X, hadn't-been-aware-of-Y' — a 'realize only when' structure that English handles with 'didn't notice until.' 寒さ is the noun form of the い-adjective 寒い, formed with the productive ~さ suffix that makes adjective stems into nouns.