But there is a path ahead. A combination of middle-range options—political reform in Baghdad, a limited use of U.S. military force, and efforts to build up local capacity and prevent new infections—offers the most hope, even if this cocktail will take months if not years to take hold.

Political reform in Iraq is the foundation on which all else rests. The replacement of Mr. Maliki by Haider al-Abadi earlier this month offers some hope that Iraq’s Shiite-dominated government might become more inclusive and convince some of the country’s minority Sunnis to turn against the Islamic State. Iran, a Shiite neighbor that backs Mr. Abadi’s government, also opposes the Sunni jihadists, which could encourage Mr. Abadi to be more conciliatory than his predecessor. But at best, we’re likely to go from abysmal to simply bad: Mr. Abadi is cut from the same cloth as Mr. Maliki and shares the same Shiite-chauvinist power base.

Still, splitting the Islamic State’s zealots off from the rest of Iraq’s Sunnis is quite doable. The Islamic State surged in June, in part, because Sunni tribes, ex-Baathists and other Sunnis had joined the fray against the Maliki government. At the height of the troop surge that began in 2007, the U.S. had turned these fighters against the jihadists. Doing so again without a significant U.S. presence on the ground will be far harder—but if Mr. Abadi’s government extends a real olive branch to its Sunni citizens, the Islamic State could rapidly lose much of its support.