How to create an instance of anonymous class of abstract class in Kotlin?

From the official Kotlin language documentation:

window.addMouseListener(object : MouseAdapter() { 
    override fun mouseClicked(e : MouseEvent) { 
    // ... 
}

Applied to your problem at hand:

val keyListener = object : KeyAdapter() { 
    override fun keyPressed(keyEvent : KeyEvent) { 
    // ... 
} 

As Peter Lamberg has pointed out – if the anonymous class is actually an implementation of a functional interface (i.e. not of an abstract class), SAM Conversions can be used to simplify this statement even further:

val keyListener = KeyAdapter { keyEvent ->
    // ...
}

Please also note this discussion about the different usage of interfaces defined in Java and Kotlin.