The provision of a room inside the school, during official school hours, is constitutionally prohibited to Christians, Jews, and almost anyone else, this going back to the McCollum decision around 1947, not even the more recent school prayer decisions of 1962 & 1963.
But, of course, a population segment that comprises a lot of voters, some of them capable of generous donations and others capable of planting bombs, does trump the Constitution ... at least in Chicago's history.
Actually, you are wrong that a prayer room is unconstitutional because well, the Supreme Court is wrong. But, it is wrong in this case because the majority religion is not the one that is being used.