I just tried this on our demo site which is using the i7media mobile kit skin. I tested with my iphone 4s. The mobile menu by design is hidden and revealed by clicking an icon in the top left corner of the page. If I login using the login module it does not remove the menu from the page, it is just hidden again as it should be until the icon is clicked again. Maybe something different is happening on your site.

Myself I'm not a fan of the login module, we implemented that because some people wanted it but I never use it on any of my sites, I prefer to use the standard login page. Any problem with the login module is just limitations of the module itself. Mobile Kit is just a skin with a special menu control, there is nothing in mobile kit that could be changed to fix or change the behavior of the login module.

My advice if you don't like the way the login module works on mobile phones, then don't use it on mobile, go into the module settings in the general tab and set Publish Mode for "Web Only", and let mobile users just use the normal login page.