TIMBUKTU, Mali — For eight days after the Islamists set fire to one of the world’s most precious collections of ancient manuscripts, the alarm inside the building blared. It was an eerie, repetitive beeping, a cry from the innards of the injured library that echoed around the world.

The al-Qaida-linked extremists who ransacked the institute wanted to deal a final blow to Mali, whose northern half they had held for 10 months before retreating in the face of a French-led military advance. They also wanted to deal a blow to the world, especially France, whose capital houses the headquarters of UNESCO, the organization which recognized and elevated Timbuktu’s monuments to its list of World Heritage sites....