Your markup is invalid – you can’t have a
div as a direct child of
ul or as a sibling of
li. There also looks to be some nesting issues.
I’m on mobile at the mo so can’t tidy up your code for you in order to test, but have a look at the source of this simplified CSS-only menu and see how it’s done there http://fofwebdesign.co.uk/template/_testing/keyboard-untabable-dropdown-menu.htm (ignore the :focus stuff for the purpose of your test case)