Brackenbury's 'River Column' and Butler's 'Campaign of the Cataracts' are what you need. 'Brack' was second in command and took over after Sir William Earle's death and Col Butler (later Lt Gen Sir William - whose wife was Lady Elizabeth the famous artist) was in command of the cavalry with the River Column. About two thirds of Butler's book is given over to how clever he is, and what a good idea the whalers were, but the last third is about events after the force reached Korti and divided into River and Desert Columns, including the Battle of Kirbekan.
Both books are effectively unaffordable in first editions but both are also available in modern reprints. Becuase I'm doing a PhD in this stuff I have spent rather more money than is sensible and own a first edition Butler and two reprints (don't ask!). If you ping me your postal address via PM I'd be happy to send you the spare reprint, gratis of course.
As to Brackenbury I think, though will have to check, that I have superseded a paperback reprint with a hardback reprint. That being so I'll happily pass on the pbk too.
In addition to the Staffs Museum try also the Black Watch who may have primary accounts by members of their regiment.
There is also a book by a guy called Gordon called off the top of my head 'My Six years in the Black Watch.' He was a corporal and was at Kirbekan. Reprints available.
Major General Andy Wauchope (KIA IC Highland Brigade at Magersfontein, 1899) was severely wounded at Kirbekan (poor feller couldn't go near a battlefield without being shot at some point in the proceedings - he was WIA I think on two other occasions). There is an old biography of him which you might pick up for 50 or 60 quid if you are prepared to go that far. I am not aware that there has ever been a reprint.
Hope that's of some help - if anything else occurs I'll supplement this info.
Regards as ever