Yep at a glance it looks like there is no styling inplace for the menu which could be your issue. I can also see there are 2 events bound to the menu items, focus and mouseout. You may be missing the mouseover event but that just speculation.
I would get your JS working and tested.
Then style your menu using the resource that @Anton provided in this answer.
Then the Flyout option, what exactly it does? I tought it makes the menu a dropdown, so it does nothing? I need to make the dropdown with js? Styling is not important if flyout makes it functional for now, but it doesn't right? Documentation is not really specific.