Removed semicolons at the end of statements, making it Swift-y.
func Close() renamed to func close(), keeping with the convention that method names begin with lowercase.
Housekeeping: minor whitespace edits
In Swift, enum members are fully fledged values in their own right. Current context doesn't require members to be Ints.
Since Swift knows variable style is of enum/type SCLAlertViewStyle, the shorter dot syntax is case statements can be used.
Default switch case indenting as per Xcode (Select block, control+I). Apple's Swift Programming Language book also follows this indentation for switch/case blocks.
Housekeeping: Spacing as suggested in the official Swift Apple text.
CGRectMake replaced with Swift CGRect(x: y: width: height:) function.
Since Swift knows labelView.text is of type (enum) NSTextAlignment, the shorter dot syntax can be used.
More cosmetic edits leveraging Swift shorthands for conciseness and better readability (according to Apple anyway!)
Using Swift's isEmpty in place of NSString's length. (countElements is another alternative). Also, if's don't have to have parentheses around them anymore (as prescribed in Apple's text)
Since duration is an optional (duration: NSTimeInterval?), Swift won't evaluate duration > 0 when duration is nil. Hence duration != nil is accounted for.
iconImageName need not be NSString, String is sufficient (even if this was a public/exposed property, Swift would import it as an NSString in its auto generated ObjC header file)
Since the switch block is exhaustive (includes all cases), default case isn't needed. In fact keeping default will let the compiler forgive any future additions to the SCLAlertViewStyle enum which will result in a viewColor and iconImageName not being set if the developer forgot to add the new case for the new enum member. Without the default, the compiler will complain causing the develop to address adding a new case in the switch block as soon as s/he adds the new member in the enum.
Removed last completion block, as it was nil. This now resembles the block in the hideView function.
Removed NSString from String constant declaration. Swift infers these variables as Swift strings automatically from it initialized values. Also noticed that kSuccessTitle were used as titles for all the IBActions. Used corresponding title string for each IBAction function.