I was recently working with a blog that featured embedded content from many social media sources. Instagram was one of them. The embeddable Instagram is great, but it doesn't naturally respond to fluid-width layouts.
Any other time I'd get that crunchy "you're doing it wrong" feeling if I styled with jQuery, but when dealing with this iframe I felt I had found an exception.
Here's an example in action as a CodePen.
Instagram embedded photos are always square, but they have a fixed-height header and footer. The plugin calculates a new height by adding the width to the heights of the fixed height elements.
I expanded a bit by allowing it to receive an options object to tweak it for various implementations. Read more about that it the Github repository.
Update January 2015
If you're using any of the old iframes, my plugin can still help. But if you get the new embed code, nothing additional is needed for fluid layouts.