Seems I managed to figure something out. I used the Part -> Attachment.. tool. Started by choosing only the edge of the sketch that I wanted to connect to the solid (and not the entire sketch). Then I chose Part -> Attachment, and chose the edge I wanted to connect to. I chose Attachment mode Inertia 2-3 (I have no idea what this means, but I tried them all), which connected my sketch to the correct centerpoint, but it was rotated wrong. So I changed Pitch to 90deg and Roll to 90deg, and that seems to be just what I need - perfect alignment.
If there's another, easier way to do this, let me know. But this seems to solve what I need.
By the way, when I say Sketch above, I mean Draft Clone every time. All the parts I'm working with are draft clones of a master sketch.
Edit: Just choose the edge to connect to, no need to select two vertices like I wrote initially.