Some combination of installing the following packages seems to have gotten it working:
libqt5quickcontrols2-5
qt5quickcontrols2-5-dev
qml-module-qtquick-controls
libwxgtk3.0-gtk3-0v5
wx3.0-i18n
Last night I installed libqt5quickcontrols2-5 and qt5quickcontrols2-5-dev, and performed a rebuild of kstars (not sure if it was necessary but I did it anyway!)
This morning I installed the list supplied by @nou which only triggered qml-module-qtquick-controls, libwxgtk3.0-gtk3-0v5, libwxgtk3.0-gtk3-0v5-dev, and wx3.0-i18n.
I should have tried running it before I installed the list by @nou but I didn't think of it. Either way, something in there has done it.
Thank you both for your help, and also thank you for the debugging suggestion @nou - that will save me a wasted night at some point soon!