Three weeks after Hurricane Sandy, fresh teams of federal disaster recovery workers rushed to Coney Island to solve a troubling mystery: few people were signing up for federal financial aid. The workers trooped into the city’s public housing towers, climbing up darkened stairwells, shouting “FEMA,” knocking on doors. What they found surprised even these veteran crews. Dozens of frail, elderly residents and others with special needs were still stranded in their high-rise apartments — even though life in much of New York City had returned to near normal. Hurricane Sandy put few agencies in the region to a more daunting test than it did the New York City Housing Authority — the nation’s biggest public landlord — as 402 of its buildings housing 77,000 residents lost electricity and elevators, with most of them also losing heat and hot water. These lifelines were cut in some of the city’s most isolated spots, like Coney Island, Red Hook and the Rockaways. An examination by The New York Times has found that while the agency moved aggressively before the storm to encourage residents to leave, particularly those who were disabled and the needy, both it and the city government at large were woefully unprepared to help its residents deal with Hurricane Sandy’s lingering aftermath.