Kim Kardashian and a former partner at Carlyle Group are launching a new private-equity firm focused on investing in and building consumer and media businesses. Ms. Kardashian is teaming up with Jay Sammons, who ran consumer investing at Carlyle, to launch SKKY Partners, they said in separate interviews. SKKY will make investments in sectors including consumer products, hospitality, luxury, digital commerce and media as well as consumer-media and entertainment businesses.