The T joint was a bit of a problem. The first thing that I did was to get a T joint from a plumbing store that fit the cross bar as snugly as I could find.
If you remember the vertical bar is a smaller diameter that fits inside the cross bar, so i made two "neck pieces" from the larger diameter tube and fit them around the smaller vertical piece.
Then I put the T joint on, which now fit pretty snugly (but still some wiggle room). To get rid of the wiggle room I pushed the vertical tube up against the horizontal T bar, drilled a hole and ran a bolt through that essentially pinned the T bar against the top of the T joint. The crossbar is still a touch loose, it twists but doesn't move laterally very easily.
The drawings are not really to scale and I am sure if you had a machine shop you could do a lot better but this was done on the cheap.