What is your issue exactly ?
All I can think about is the fact that you never save the new value of level.
So likely are missing an action "Local Storage - Set Item" in your Event Sheet 2 before the "Wait" action.
Set the key to "levelKey" as this is the key you load in your menu layout and set the value to the global variable "level".
Instead of a wait action, make an event "Local Storage - On item set" "levelKey" and in that event go to the menu layout.
Your even sheet 2 should look something like that :
As for the compatibility with APK, I must admit I don't know if Intel XDK or Cordova have yet integrated support for Local Storage.
I know it works on Chrome for Android and Firefox for mobile, but I don't know about "native" applications. (They should as they are based on chrome though)