When you use background-size: cover the background image will automatically be stretched to cover the entire container. Aspect ratio is maintained however, so you will always lose part of the image, unless the aspect ratio of the image and the element it is applied to are identical.
I see two ways you could solve this:
Do not maintain the aspect ratio of the image by setting
background-size: 100% 100% This will also make the image cover the
entire container, but the ratio will not be maintained. Disadvantage
is that this distorts your image, and therefore may look very weird,
depending on the image. With the image you are using in the fiddle, I
think you could get away with it though.
You could also calculate and set the height of the element with javascript, based on its width, so it gets the same ratio as the image. This calculation would have to be done on load and on resize. It should be easy enough with a few lines of code (feel free to ask if you want an example). Disadvantage of this method is that your width may become very small (on mobile devices), and therfore the calculated height also, which may cause the content of the container to overflow. This could be solved by changing the size of the content as well or something, but it adds some complexity to the solution/