It took more than a week to get over the cold.

Literal

Cold [object-を] cure [purpose-のに] one-week more-than took.

一週間以上かかった — 以上 means 'more than/over.' のに here marks the goal of the duration (curing the cold) and かかる ('to take [time]') gives the elapsed amount. A casual past-tense report of a stretched-out recovery.