Mount Everest is not located in any city. It stands on the international border between Nepal and China (Tibet Autonomous Region) in the Mahalangur Himal subrange of the Himalayas.
27.9881° N, 86.9250° E — precisely on the Nepal-China border. The summit ridge defines the boundary.
Nepal: Sagarmatha National Park (UNESCO), Khumbu Pasanglhamu Rural Municipality
China: Qomolangma National Nature Reserve, Tingri County, Shigatse Prefecture
Physiological limit: permanent human habitation above 5,500m is unsustainable (hypoxia, cold, UV). No agriculture, no water infrastructure, no supply chains. The "city" supporting Everest is Kathmandu — 160 km away, 1.4M people, expedition hub.
Everest is a border landmark, not an urban one. The nearest city is a week's walk or a flight away.