This is the bit of header.php you're interested in:
<a href="http://hildurgunnlaugs.is/" title="Hildur Gunnlaugs" rel="home">
<h1 id="blog-description">arkitekt FAÍ</h1>
One thing you might try is to insert your image immediately after the opening tag of the
branding div, or after the opening of the
blog-title div, depending on which looks best. You will also need to add a style for
div#branding img (or
div#blog-title img) if you need to float the image.
Don't worry about gaps and stuff. We can sort that out with CSS: or, at least, I'd be surprised if we couldn't.
Once again, I would exhort you to use a child theme. They're trivially easy to create and can save you a lot of grief later on.