These inspectors general are removable not by the president but by the bodies that appoint them—for example, by a unanimous vote of the Capitol Police Board. Yet all have a vote in the election of the Cigie chairman, who oversees the council’s management.

Cigie’s structure raises constitutional problems because of its investigative power. Probes into inspectors general and their staffs go through Cigie’s six-person Integrity Committee. Four of the committee members are inspectors general appointed by the Cigie chairman, one is an official from the Federal Bureau of Investigation, and one is either the director of the Office of Government Ethics or someone he designates. None of these officials are appointed to the committee or removable from it by the president.

The Integrity Committee thus has the power to investigate members of the executive branch, issue damaging reports on them, and recommend sanctions, including termination. The findings are sometimes published before the president makes a decision, causing injury to their targets. Yet the committee is unaccountable to the president or anyone else.

The problem has come to a head because Mr. Trump’s recent terminations of inspectors general mean a higher portion of Cigie’s membership currently consists of nonpresidential appointees. On Jan. 1, Hannibal “Mike” Ware, who in 2018 was appointed inspector general of the Small Business Administration by President Trump and confirmed by the Senate, began his two-year term as Cigie chairman. But on Jan. 24 Mr. Trump dismissed him. The vice chairman, U.S. Postal Service Inspector General Tammy Hull, has been acting chairman ever since. But Ms. Hull wasn’t appointed by the president and owes her position to the USPS Board of Governors.

The chain of command from the president to Cigie has always been highly attenuated, but at the moment it’s utterly severed. The president can’t remove Ms. Hull, and she isn’t supervised by any officer of the executive branch. But she is now empowered to carry out all the executive functions of her position.

Kimberly Howell, the current chairman of the Integrity Committee, is in a similar situation. She is the inspector general of the Corporation for Public Broadcasting, which by its charter is not “an agency or establishment of the United States Government.” Thus, she too isn’t appointed by the president or overseen by anyone in the executive branch. Yet she has the authority to investigate some officers in the presidential chain of command.

The president can’t solve this problem alone because Cigie and the Integrity Committee elect their own leaders. So the country is stuck with a body that can do enormous damage to the executive branch while remaining completely outside the executive’s control. Such a structure can’t survive under the Constitution. It is time for Cigie to go, and for the president to accept no recommendations from it until it does.