EMB-145 Flight Attendant reported high cabin air temperatures resulting from one pack inoperative during flight.

Narrative

Aircraft X obtained earlier at ZZZ at/about XA20 during scheduled swap; crew delayed onto aircraft due to earlier delay originating flight. Advised by captain MEL for air conditioning system/one pack only in operation. Aircraft became progressively hotter through the day; last leg flown on low enroute course to ZZZ1; unable to cool to 'normal range' at designated cruise altitude; front area by FA jump seat recorded 85-90 degrees on small handheld thermometer provided by company during 75% of flight. Passengers were quite uncomfortable but unusually forgiving of temperatures; FA monitored for any heat-related stresses; especially the elderly. Despite conditions; crew operated aircraft successfully ZZZ-ZZZ1; at arrival ZZZ1; captain deemed aircraft 'unfit for continued operation'(sic).Cause: Bad air conditioning system pack deferred/Minimum Equipment List. Cabin conditions exacerbated by outside ambient temperature at outdoor gate; boarding door open to ramp temperatures/ reflected heat from operational surfaces and equipment