Engels claimed on Twitter that Mugabe attacked her while her guards looked on and did nothing to help her.
"What is a girl compared to a woman beating you and 10+ bodyguards standing back leaving her to do this s***," she wrote.
"She split my head open in three places with an extension cord and used the plug to hit me."
Engels's mother, Debbie Engels, said via social media that she was going to the police to lay charges of assault against Grace Mugabe.
Grace Mugabe's spokesman has not commented on the incident.
The Mugabe sons used to live in Dubai in a palatial house rented by their parents, but had to leave suddenly, allegedly in connection with incidents involving drug or alcohol abuse.
They are studying in Johannesburg and are well known at various luxury nightspots, where they are reportedly generous with their cash.
Both were recently evicted from their luxury Johannesburg apartment after they were caught up in a drunken brawl in which a security guard was injured.
Garnet Basson, CEO of the Capital Hotel group, declined to name Grace Mugabe but confirmed that guests were asked to leave a hotel after an "incident".
"We did have the guests removed from the premises due to a scenario where we thought it was not good for them to stay there," he said.
