If it's still using CPU it sounds like it's still doing something, which might be hanging before the window can be displayed. We've had various bugs in the past that might cause this. Up until this past fall, the most common cause was an issue with authenticated proxies, but since that was fixed it could be anything.
To find out what it's doing, you should look at %LocalAppData%\Airspace\log.txt and paste the tail end of that here.
If you want to rerun and generate a new log, be sure to terminate off both LeapMotionAppHome.exe processes first. It cannot run more than a single instance. If you neglect to kill off the previous processes, the new log won't show much.
If needed, there is a way to get a detailed node-webkit log, but hopefully won't be necessary. If one LeapMotionAppHome.exe is using CPU it's likely doing something sufficient to be visible from the main log.txt