Khartoum fell in January 1885 because Britain refused to finance a relief campaign.

Evidence

The Egyptian government was powerless to send in more reinforcements, and neither Lord Cromer nor the British Parliament was inclined to bankroll a largescale campaign into the Sudan against the Mahdï. After months of siege, in which the food supplies grew ever shorter and conditions ever more miserable, the soldiers of the Mahdï stormed the city in January 1885. Charles Gordon was killed near his bed chambers. Ibrahïm Fawzï was severely beaten and taken prisoner by the Mahdists. — Eve M. Troutt Powell, A Different Shade of Colonialism_…, 532