I think a strong argument can be made that de facto school segregation has never ceased to exist, but I assume you're asking about the last legally segregated schools.
To that end, I think the first thing to realize is that while Brown v. Board of Education came down in 1954, court supervised desegregation efforts stretched well into the 2000s and a few continue today, so in a sense efforts to end de jure segregation (that is, the effects of legally sanctioned segregation) continued well into the era that is off-limits for this subreddit (e.g. topics that happened in the last 20 years).
In other words, I don't know if your question can fully be answered within the bounds of this sub. Schools were not simply segregated one day, and then desegregated the next; rather they were put under court-ordered supervision plans and the process is still technically on-going. But I'll mention a few major milestones that occurred pre-1996.
First off, Brown did not detail much of a timeline for when schools needed to desegregate, the Court simply said "with all deliberate speed," and schools in the south not only didn't begin responding for a decade or more, they actively fought desegregation. Hence the Supreme Court returned to the issue in Griffin v. County School Board of Prince Edward County (1964), and later Green v. County School Board of New Kent (1968), and Swann v. Charlotte-Mecklenburg Board of Education (1971).
By 1971, however, schools in the south reached similar levels of segregation as the rest of the country. Which is to say that "only" 20 percent of black students attended all black schools.
To achieve this, courts had implemented court-supervised desegregation plans. In 1991, however, the Supreme Court ruled that these plans were never meant to be permanent, and once schools had made a "good faith effort to desegregate," they could be released from the court order. See: Board of Education v. Dowell, 498 U.S. 237 (1991).
In 1990, there were still more than 400 school districts under court-supervised desegregation programs. This number dwindled, but as mentioned above, many of these court-ordered plans persist today. Unfortunately, I don't have a solid number on how many still exist, but that would probably not be appropriate for this sub anyhow.
Sources:
First, the cases quoted provide a great deal of information. Otherwise, this 2011 paper, "Brown Fades: The End of Court-Ordered School Desegregation and the Resegregation of American Public Schools" published by the Journal of Policy Analysis and Management provides a pretty good analysis of court-ordered plans between 1990 and 2009, and provides the exact numbers quoted in this post.
Desegregation was still actively going on in Duval County, FL (Jacksonville) until 1990, when the NAACP and Duval County Public Schools agreed to end forced busing, the last active vestige of desegregation.
The NAACP continued fighting legal battles to continue desegregation in other ways. But this appeals ruling in 2001 states that by 1990, forced busing was doing more harm than good.
Side note: this ruling shows desegregation was not legally settled in the 12th-largest US city until at least 2001.
In reality, the NAACP and others still to this day work diligently to ensure desegregation is a reality in the South.
Other school systems in Florida are still under active federal desegregation orders.
Having worked with one of the school districts cited in the article (and from reading the article itself), I believe this is more of an administrative oversight that was never corrected than active desegregation.
But in short, the answer to your question is, school desegregation is still an issue, but active desegregation efforts that directly affect students were largely finished by the early 1990s.