How can I use the Knife tool to create a line between two vertices?
When I try to use it, the tool simply deselects my first vertex.
If I read this correctly, you don't really need the knife for this, you can just select the 2 verts and press j. This also works with verts separated across longer distances. The join (j) line will cut a vert into every edge it crosses in a straight line between the selected verts.
If you're using Blender 2.79, and you right-click instead of left-click, you cancel the operation. use the Left mouse button to create each vertex then press Enter to finish off.