responsive menu issue

I have been looking for a simple mag theme to use as a template for a health department website. So far, this looks great, but I am having problems with responsiveness of the menu. When I shrink on other devices, I see the hamburger but it does not kick in. The menu just collapses to several lines, which I don’t like.

I also would like to put a small logo to the left of the header text. Is there any way to do that other than just making the whole header an image? I have struggled with putting some background color in the heading.

Here is my little draft: http://kas.website/lhd2.

Hope you can help remedy because otherwise I think it will work well.