NEW YORK — After 7-year-old Nixzmary Brown was discovered emaciated and beaten to death despite warning signs of her abuse, sweeping reforms were made to New York’s child welfare agency. Millions were invested, caseworkers were added and police presence beefed up. But the legal system that deals with troubled families remains stagnant in the two years since Nixzmary’s death, choked with a backlog of hundreds of cases that forces children around the state to languish in foster care. ‘‘You have a certain exigency to cases involving children,’’ Judge Ann Pfau, chief administrative judge for New York state courts. ‘‘You want to be able to resolve problems as quickly as possible, you just want stability for the child. Six months in a criminal court case is different from six months in a family court case. Six months is a very long time for a child. We think about it all the time.’’ There are 47 judges assigned to Family Court in New York City, and about 146 statewide who handle more t