This is a reference to an incident recounted in 1 Samuel 16.14-23 in which Saul, the first king of Israel, is beset by an evil spirit sent by God. His servants hatch a plan to cure him by hunting down the most talented harp player in the kingdom. They eventually light upon David, a shepherd of the tribe of Judah. He is duly summoned before Saul and when he plays the evil spirit departs.
This reference, which sees the governess play the part of the possessed and Miles of the exorcist, is one of many hints that it is she, and not her young charges, who has fallen foul of diabolical influences.