1. What does it mean?
왤케 compresses 왜 이렇게. It asks why something is a certain way, but can also work as an expressive exclamation about how strong a quality feels.
Also written or said as 왜케, waelke
2. The nuance
It can introduce praise or a complaint. In formal writing use 왜 이렇게; the clipped spelling alone does not tell you whether the speaker is angry.
3. Examples
“오늘 하늘 왤케 맑아? 사진 찍고 싶네.”
Why is the sky so clear today? I want to take a picture.
“가방이 왤케 무겁지? 책을 너무 넣었나?”
Why is my bag so heavy? Did I pack too many books?
Sources
References support the meaning and usage. Example sentences are written for NoCapWiki.