As you found out – if you go to appearance -> customise – then there’s some more options for editing the header layout.
In your case I would add the name and description back and then use css to hide them and position the search box as you want.
Are you happy with how to do this (using display:none;) or would you like some tips? I don’t know what your css skills are like.
To be able to customise the css I would recommend the Jetpack plugin as it has a built in css editor – along with a bunch of other helpful features (you can disable anything you don’t like).