I agree with the comment of @sempaiscuba
In 1871 there we no railways to Kimberley. The town of Kimberley was established in 1878. It become part of the Cape Colony in 1880. The railway to Kimberley was only completed in 1885. I assume the planning of the railway started shortly after the diamonds were discovered there in 1871. (Kimberley history)
The nearest large town/city is Bloemfontein. That is about 150 kilometers away if you walked. With that being said the railway to Bloemfontein was only completed in 1889. I believe that the railway to Kimberley was of greater importance than the one to Bloemfontein. Also important to note that Kimberley was part of the Cape Colony and Bloemfontein was part of the Orange Free State. Kimberley was under British rule and the Free State was an independent republic. (Bloemfontein railway) (Colonial history Bloemfontein)
King William's Town is about 550 kilometers from Kimberley if you walk. Depending on rivers and mountain ranges this could be closer to 650km. For a miner to travel form King William's Town to Kimberley could easily take more than a month (this is under the assumption that someone can travel 15km by foot per day).
The chances of a miner having an ox-cart would have been very small given their living standard. If they were lucky enough to have a horse this journey could be shortened by about two weeks (assuming that a person can travel 35km per on a horse). (Travel times)
It is important to bear in mind that the miner would have the Orange river among other rivers to cross. There might have been some bridges, but these would be far apart. Other challenges include the scarcity of water. While there would be small towns on the way to Kimberley, there would probably not be a lot of water in between apart from a few rivers. This is especially true the closer you get to Kimberley. The landscape close to Kimberley is considered by many South Africans as semi-desert.
Other options could include taking a coach for Port Elizabeth, but I doubt that any miner would be able to afford it. These miners often went away from home in order to make money that they could take back home to care and provide for their families.