Hi there, Toby! Thanks for getting in touch and welcome to the forums. Sorry you’re hitting some trouble here, but hopefully we can figure it out here together. 🙂
That’s really odd and definitely not what should be happening. I went ahead and set up a test site running the default Twenty Fourteen theme like you (thanks for doing that, by the way!) to see if I could replicate the issue, but was unable to. The mini calendar (and other event widgets) appear to display correctly when I give it a shot.
Here’s a screenshot of my widget settings and a screenshot of the mini calendar in my sidebar. Do your settings look similar? Is it possible that sidebars have not been registered in your theme(s)? Perhaps the widgets should go in a different sidebar?
Let’s start there and please let me know what you find.
Cheers!
Geoff