I have no real historical knowledge of the region, and very little about the time period more broadly, but what little I've read about Kilwa was fascinating. Any suggestions would really be appreciated! Cheers.
Neville Chittick conducted a dig at the stone town of Kilwa Kisiwani back in the 1970s, and his book Kilwa: an Islamic trading city on the East African Coast contains a chronology of the city and his interpretations of the archaeological finds his work uncovered.
Something more recent would be this article by J.E.G. Sutton which attemts a critical examination of the Kilwa Chronicle by comparing its narratives with archaeological evidence.
Timothy Insoll also dedicates about 70 pages to Kilwa and other Swahili states in his book the Archaeology of Islam in Sub-Saharan Africa
As you might expect, the Swahili states loom large in the study of Islam in Africa. Roman Loimeier dedicates significant attention to the Swahili states in Muslim Socieites in Africa: a Historical Anthropology. As do Nehemia Levtzion and Randall Pouwells in The History of Islam in Africa, and David Robinson in Muslim Societies in African History.
If you are less interested in academic perspectives, but would prefer historical first-hand accounts, you could trawl through this collection of primary sources. If you are particularly focused on Kilwa, I would recommend starting to look at sources after about 1200 AD. Before that, Mombasa was the major player on the East African coast, and Kilwa is unlikely to have drawn particular notice.
Because i've seen /u/Commustar answer things like this I'm putting his name here.