I began the installation in the highest corner. Pulling the layout from the opposite wall, I locked the first new joist in position with a scrap screwed to the nearest old joist.
With the high point set, I measured the height from the joist to the laser, then shimmed up the other end of the joist until the measurement matched. At the midpoint of the joist, I measured the gap between the joist and the top plate of the supporting wall below and cut a block to fit. The first joist set the height for the rest of the floor, so it was just a matter of working my way across the floor, bringing each joist up to that level.
The exterior wall was framed at 16 inches on-center except under the windows, so I was able to use the studs to set the rough layout for most of the joists. The studs also gave me a place to attach the joist ends. For each joist, I checked the layout at both ends, and if they didn't land next to a stud, I added shims or blocking to the nearest one.
As with the first joist, I measured and cut a block to fill the gap between the bottom of each joist and the wall below. After slipping the block under the joist and screwing it to the plate ...
To stiffen the floor, I glued and screwed solid blocking between the joists in three separate spans across the floor, and I blocked every space between the new joists and the old joists.
At the ends of the joists, I added horizontal 2x4 blocks between the wall studs to close off the bottoms of the stud bays and to provide nailing for the wallboard.
I needed to provide attachment for the subfloor along the walls running parallel to the joists, but it would have been wasteful to run a full LVL joist where I just needed a "nailer" below the studs. Instead, I screwed vertical blocks to the beam that forms the plate every 16 inches to catch the edge of the OSB subfloor.
Finally, I insulated the ends of each joist bay against the outside wall with fiberglass batts. To complete the floor system, I glued and screwed 3/4-inch OSB subfloor to the joists, staggering the end seams as I went along (18). In the end, the extra time I spent paid off. The floor was completely flat, level, and solid as a rock—a perfect foundation for the new kitchen cabinets.