At what latitudes does daytime last exactly 24 hours once in the year?
The latitudes today are 66°33′46.1″ N and S, the Arctic and Antarctic Circles.

By definition, north of the Arctic circle, and south of the Antarctic circle, everywhere gets at least one day with no sunset, which I guess is what you mean by 'exactly 24 hours of daylight'? 

From the linked article: 


  North of the Arctic Circle, the sun is above the horizon for 24 continuous hours at least once per year (and therefore visible at midnight) and below the horizon for 24 continuous hours at least once per year (and therefore not visible at noon).