Add Google custom search box

Hello! I have added a Google custom search box in my header in order to search on a WordPress network site by following this guide , but it does not look good. I have tried to use Firebug to style it with custom css in my child theme, but I have still not managed it jet. I would appriciate if someone had a an advice about how I can make the two white areas overlap, remove the border and make the search button visible.I managed to style an ordinary search box nicely, but the ordinary search function in WordPress is only able to search within this site, not the network. Therefore I tried this solution, but maybe someone has a better solution?