Very sorry to hear her mom is ill.
She should definitely apply for her visa in advance next time. No one wants to spend 10 hours on a flight just to be turned around and put on the first flight home!
She will need to show strong ties to the US in order to be granted the visa (a job to return to, a home that she is maintaining while traveling, plenty of money to support her trip, health insurance, etc.).
The rule of thumb is that she should spend as much time out of the UK, as she was here for. It may be difficult for her to be granted a visa so quickly after her previous entries. But you don't know until you try!
