For how long is a sprained ankle swollen?

Depends: It depends on the severity of the sprain and how it is treated.

Long time: Months to a year.

2 weeks to a year: A sprained ankle swells initially because of the injury to the soft tissue and the blood/lymph vessels torn at the time of injury. It takes a while for it to resolve. Anywhere from 2 weeks to up to a year. If you have a tendency to have swollen ankles for other reasons it may persit forever.