My mother caught a cold and couldn't smell anything.

Literal

Mother [topic-は] cold [object-を] caught-[sequential-て], smell [subject-が] could-not-smell

風邪をひく = 'to catch a cold' (ひく literally 'to pull/draw'). 嗅げなかった is the negative potential of 嗅ぐ (to smell) — 'couldn't smell.' におい (smell) + が (subject) — the ability pattern marks the object of ability with が.