A quick update.
It almost works as intended with this code:
$target = $(e.target),
$draggable = $(e.draggable.currentTarget);
The only problem now is that there is no 'filter' option for kendoDropTarget(). So the whole UL becomes a drop target, as well as all its children (including LIs, which is good, but also including anything inside those LIs, which is bad).
I was thinking about traversing the tree but I can't be really sure what will become the $target as it depends on where user drops the draggable content. It could be the UL or LIs or anything inside them. Any way to just tell kendoDropTarget to only accept drops on LIs (or anything inside) and then append stuff after the LI?