Hi honeyelle,
I looks like these lines in your theme are making the larger menu appear only when your site is displayed on sites larger than 600px
481 @media screen
482 and (max-width: 600px) {
Therefore if you changed line 482 to say 300px it should show the menu on most devices. You may need to play around with the number to see which works best.
That way the small menu will only ever appear on devices with width below there. Even mobile devices tend to have widths greater than 300 pixels these days.