Swift: How do I return a value within an asynchronous urlsession function?

You should add your own completionHandler closure parameter and call it when the task completes:

func googleDuration(origin: String, destination: String, completionHandler: (Int?, NSError?) -> Void ) -> NSURLSessionTask {
    // do calculations origin and destiantion with google distance matrix api

    let originFix = origin.stringByReplacingOccurrencesOfString(" ", withString: "+", options: NSStringCompareOptions.LiteralSearch, range: nil);
    let destinationFix = destination.stringByReplacingOccurrencesOfString(" ", withString: "+", options: NSStringCompareOptions.LiteralSearch, range: nil);

    let urlAsString = "https://maps.googleapis.com/maps/api/distancematrix/json?origins="+originFix+"&destinations="+destinationFix
    println(urlAsString)

    let url = NSURL(string: urlAsString)!
    let urlSession = NSURLSession.sharedSession()

    let task = urlSession.dataTaskWithURL(url) { data, response, error -> Void in
        if error != nil {
            // If there is an error in the web request, print it to the console
            // println(error.localizedDescription)
            completionHandler(nil, error)
            return
        }

        //println("parsing JSON");
        let json = JSON(data: data)
        if (json["status"].stringValue == "OK") {
            if let totalTime = json["rows"][0]["elements"][0]["duration"]["value"].integerValue {
                // println(totalTime);
                completionHandler(totalTime, nil)
                return
            }
            let totalTimeError = NSError(domain: kAppDomain, code: kTotalTimeError, userInfo: nil) // populate this any way you prefer
            completionHandler(nil, totalTimeError)
        }
        let jsonError = NSError(domain: kAppDomain, code: kJsonError, userInfo: nil) // again, populate this as you prefer
        completionHandler(nil, jsonError)
    }
    task.resume()
    return task
}

I’d also have this return the NSURLSessionTask in case the caller wants to be able to cancel the task.

Anyway, you’d call this like so:

googleDuration(origin, destination: destination) { totalTime, error in
    if let totalTime = totalTime {
        // use totalTime here
    } else {
        // handle error     
    }
}