?」「

'Is Aki off today?' 'Sounds like she let her cold get worse.'

Literal

'Aki-[affectionate-ちゃん] today off?' 'Cold made-worse [seems-みたい][confirmation-ね]'

こじらせる means 'to aggravate, to let worsen' — 風邪をこじらせる is the classic collocation. みたいね is casual inference: 'it seems like.' The first speaker drops all particles (アキちゃん今日休み?) — natural casual speech pattern.