Just saw PBS' 'Diary of Dr. Livingstone' and it seems crazy that the source hadn't been found.
The easiest way to find the source of a river is to simply follow it upstream, but the Nile passes through an enormous disease infested swampland that has few distinct channels, and is still essentially impossible to travel through, called the Sudd, making that impossible. So to find the answer, you have to come at it from up-stream, but the hydrology of the Great Lakes region of Africa is extremely complex with many rivers, so figuring out which rivers connected which lakes and how was extremely difficult. The source of the Nile is still a difficult question to answer, and it depends on semantics as much as on geography. There were, of course, 100 other challenges to mapping continental Africa, but even without them, it's a difficult question to answer.
The books "The White Nile" and "The Blue Nile" are very readable accounts of the efforts to find the respective sources. The sources had been a point of speculation since ancient times. Their very long routes through inhospitable country, including "darkest Africa" were especially challenging, and rewards (being the solving of a point of geography) were poor. Speke and Burton's particular travails in the 19th Century underline the problems following the river presented to anyone -disease, difficult terrain and hostile indigenes abounded.
The violence of the Arab slave trade in East Africa at the time was a big hindrance to the Victorian era explorers. The whole region was dangerous. There were many raids and massacres at the time. Livingstone witnessed a massacre of an entire village during his travels.
African chiefs were rightfully distrustful of any outsider, believing them to be agents for the slave traders. And they had a point, since many of the explorers depended on the Arabs for for supplies and porters.
See:
Or a recent book on the topic: Explorers of the Nile The Triumph and Tragedy of a Great Victorian Adventure
TIM JEAL