The flight crew’s delayed landing after an in-flight engine fire, which reignited after both fire bottles were discharged and resulted in substantial damage to an engine pylon. Contributing to the delayed landing was likely the flight crew’s perception that the fire had been extinguished due to the disappearance of the primary engine fire warning indications after the fire detection loops were damaged and that a landing as soon as possible was not perceived to be necessary. Contributing to the duration of the fire was the contamination of an engine fire isolation system component which resulted in hydraulic fluid leaking into the designated engine fire zone after the engine was shutdown and the fire button was pressed.