I see what you did there, and i guess i am having trouble getting my point across.
the entire site, or at least the body of it, never gets any larger than 1140px wide, at least with whatever current settings she has now. From browsing the code it seems that the header image will display at a maximum of 100% of its parent container, (.site-branding) which is 1090px wide.
So it’s safe to assume that an image any wider than 1090px (and with any height) would be great! otherwise whatever she puts in there will display at it’s actual dimensions, i.e. not be scaled larger, thus pixelating it.
All that led me to believe it’s not the image size of her current image that is the issue, but rather the quality of the current image.
Which is why i asked how the image was created as well as how it was exported/saved. those affect how an image can appear on a screen.
and again i will state that i really don’t think anyone from the theme can help with an images initial quality…
i am a graphic designer and front end web developer. i have a very firm grasp of this current situation