Why didn't the North do anything about the segregation in the South?

The simple answer is that white Northerners either agreed with the notion that blacks were inferior to whites or were indifferent to black people.
Jim Crow, a nickname for the South's legal (or, de jure) system of segregation, comes from the name of a minstrel figure invented in the 1830s by the actor and playwright Thomas Dartmouth Rice. Minstrelsy was popular in both the North and the South at the time. "Blackface"—the practice of white and black actors appearing in exaggerated, darkened makeup created from burnt cork—declined in popularity after Al Jolson's The Jazz Singer in 1927. However, some of the tropes popularized by minstrel actors—bug-eyed, easily frightened characters, Aunt Jemima types, and slow-witted characters—were roles that were still handed to black actors well into the post-World War II era.
I mention all of this to say that the North and the West also had racial prejudices, though black people in the North did not encounter some of the mortal dangers, such as lynching, that they endured in the South. The North, like the South, benefited from segregation, and in the North de facto segregation existed to keep jobs, property, and social prestige only among whites. Ghettos, such as Harlem, existed because whites did not want to share capital or employment opportunities with non-whites—particularly black people.