It has been just over a year since I applied for my second one.
I don't remember if you need proof of her birth, but you do need details of her birth - birth names, birth date, birth place etc. I think they trace it on their side.
Then you need proof (SA birth certificate) that one of your parents was descended from your grandmother, and then proof that you are your parents son.
Don't pay an agency - waste of money.
I think the visa will cost you about R8000 and the NHS fee about R40k these days. The NHS fee is what you pay to go onto the NHS as soon as you arrive.
You are also advised to provide proof that you are looking for work, or capable of looking for work, when you arrive. So, letters of you applying to employers, that sort of thing. They don't want you to come here and then sponge off the state. You aren't allowed to anyway, but they don't want to be put in an awkward position of having to look after you.
You also need some cash reserves to prove that you can survive until your first paycheque. Or, proof that someone will support you for a while until you are setup.
One more thing to note about the way it works. You put the date that you want to arrive. Let's say you want to arrive on June the 1st. Make sure you specify that date. You then get a 30 day visa that starts on the 1st of June. 30 days only. From the date that you arrive in the UK, you then have 10 days to go pick up your biometric residents card at a place of your choosing (has to be a post office or police station, but in a town/city of your choice).
When you arrive in the UK, you bring your passport with that 30 day visa (called a vignette) and a letter containing details of your biometric ID card. They will let you into the country, and you then have to fetch that card within 10 days.
Feel free to ask any questions you like. I went through this process in December 2018.
EDIT: More detail, because I'm thinking of it.
The application process is not too bad. You fill in a LONG form on the website. Super long, with lots of detail about your family history, visa history, criminal history etc.
Then you arrive at the processing centre. They will scan your documents and attach them to your application (digitally). They won't bother scanning anything they don't need, as far as I can remember. Make sure you tick all of the boxes and you will be out of the processing centre in 15 minutes excluding wait time (which is not all that long either).