I just needed the seat itself, so I cut off the legs and back using my Japanese saw. This saw is perfect for this type of job because it's very flexible so it can be bent to cut fairly flush.
Next I sanded it to strip the old varnish.
There were a few gaps which I fixed with high performance wood filler,
Once the surface was nice and smooth I finished the seat with 2 coats of primer and 4 coats of spray white paint, and 3 coats of polyurethane. I sanded the surface with 400 grit paper in between most of these coats, except the paint.
The legs I made out of some 2x6s I had laying around. I planed them, then cut them to be square, sanded them and cut a bevel with the hand planer on the corners.
I used maple dowels for the joints. I didn't use a self-centering jig for these. Instead, I made a marker out of one of the dowels with a screw in it, then used that to mark the spot where the join would go. Then I drilled all the holes with the drill press.
Once the joints where glued together, I used my home made squares to clamp them square
This is how the legs looked like when drying up after joining both sides. I ended up not being able to square all sides so the legs are a tiny bit crooked. I need to make more of these squares.
I added some reinforcements in the last minute to make the legs stronger.
Finally, after a coat of stain and two coats of shellac, the finished product (with its matching desk):