I have exactly the same problem with using WordPress 4.01 and Responsive Menu plugin version 2.2.The weird thing is when you resize a desktop browser to a width-size like a tablet (no text visible, only small lines) it show options on a left-hand side, see example on https://wordpress.cynicsteaparty.com. When user extend browser to the point when text is visible – second level become invisible and menu becomes irresponsive. The Responsive Menu plugin version 2.1 doesn’t have this problem and works as expected for the desktop.
Any idea what need to be changed to make it work?