Okay, so I have found several mistakes.
First, you were checking for the LocalStorage data at the start of the "Game Over" layout. The data from LocalStorage is best to be updated at the start of the game (the first layout):
Second, here you were setting the "high_score" item to a string "TotalDistance", I've done this:
Third, you are reseting ALL global variables, including the high score, at the start of the "Game" layout. That way, you are actually erasing the highscore data which you need to display later.
Instead of reseting all global variables, simply return back to 0 those that must be reseted, as an example I have set CurrentDistance to 0, but I guess there's more that need to be reseted:
So, here's what I've got now:
30 is the highscore, while 24 is my new score..
Hope this helped
edit: in the first image, the 'on item missing' event is not really necessary, so delete it..