Hi guys, this is my first post and pretty new to the furniture restoration realm. I want to look into options on fixing the basket weaving spindle on this chair. I found it on the street a couple years ago. Cant find any marking of designer or manufacturer either so if anyone has an idea I would love to know. Either way I love it so much and wanna make it as sturdy as new. 
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I don't know who designed or produced the chair. As for restoring it--you mean the sheet cane back, right? It looks very dry and I think replacing it competely is the only viable way to go. It's about the simplest cane repair to do, but you will need to invest in a few inexpensive tools to do it well.
Sheet cane comes pre-woven in sheets and you can get various styles of weave, including the one that's on the chair now.
You need a cane chisel to remove the spline that holds the sheet of cane in the groove. It's possible to use a screwdriver or wood chisel but a cane chisel works better and is less likely to damage the wood at the edge of the groove. I have used all of the above and the cane chisel is definitely the way to go.
You will need a set of 8 or so wooden wedges that are made for replacing sheet cane, and it's best to use the glue that is made for sheet caning.
Other tools include scissors, hammer, box cutter or very sharp wood chisel, and measuring tape or yardstick, all of which you probably already have.
And lastly, you have to buy the sheet cane itself, plus enough of the right width of spline to go around the perimeter.
There are lots of videos and instructions online on how to replace sheet cane so I won't go into great detail here. But basically you gently chisel out the spline and remove the old cane. Clean out the groove completely. Cut the new sheet cane so that it is 2" bigger on each side and soak it for 20 minutes in warm water. Center it over the back of the chair and gently push it into the grooves using the wedges and a hammer. Alternate sides and make sure the weave is square to the chair frame. Don't try to get all the slack out; that will disappear as the cane dries. If you try to get it too taut, the cane will break.
When the sheet cane is pushed all the way into the groove and is staying put, apply a line of glue in the groove and add the spline. Cut neat diagonal corners so that it fits exactly. Trim the excess sheet cane away with the box cutter. (I usually do this before I put the spline in because I don't like it to show it all, but that's a little harder to do.)
Let the cane dry completely before you sit in the chair.
There are a bunch of places that sell sheet caning supplies online so you should have no problem finding what you need.
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