Did you get a chance to read the FAQ at the link above? It explains all this. And it has a link to the right edge Driver . 
But the main point is that you can’t trigger an Alexa routine with any switch, virtual or physical. That’s an Amazon decision. Their routines can only be triggered from a sensor or a lock.
So…from the SmartThings side, The solution is to use a virtual device which is BOTH a sensor and a switch. When you turn on the switch, either by voice from Alexa or with a SmartThings routine (not an Alexa routine), in the sensor looks like it opens. When you turn off the switch, the sensor looks like it closed. So it’s easy to control, but you can still trigger an Alexa routine with it. 
Details in the FAQ:
FAQ: Can I trigger an Echo Action without Speaking to It?
The link to the correct edge driver is at the end of the first section of the FAQ, right above the “old information“ section. It’s also in the wiki how to article linked to in the FAQ.
Also, in your case it’s important to understand that your integration will only be one way. You will be able to use smartthings to turn the WiFi switch on and off using this method, but because an Alexa routine cannot be triggered with a switch, you will not be able to use your Wi-Fi switch to trigger anything in SmartThings. I don’t know if that will meet your use case or not.