1. What does it mean?
어케 is a casual written form of 어떻게. It can ask for a method, express surprise that something happened, or appear in an anxious question about what to do next.
Also written or said as eoke
2. The nuance
어캐 is also encountered informally, but 어떻게 is the standard full form. This is different from 어떡해, which contracts 어떻게 해, although their uses can overlap in quick messages.
3. Examples
“이 파일은 휴대폰에서 어케 열어?”
How do I open this file on my phone?
“막차 놓쳤는데 이제 어케 돌아가지?”
I missed the last train; how am I going to get back now?
Sources
National Institute of Korean Language
References support the meaning and usage. Example sentences are written for NoCapWiki.