First off, you should apply the scale of your objects and curves with Ctrl
Ctrl + A
A. If the scale is not applied, it will mess up your modifiers and materiels.
Now, creating an entire weave of cord around your chair would be fairly difficult, and would probably be best solved by a crazy Geo node setup. However, if you're just wanting the wrapping in your file to better fit the object its wrapped around, here's one way:
In the modifiers of your curve spiral, change the offset to zero. Then, at the top, with the curve selected, under Object > Convert, select Mesh. This allows you to apply materiels and other modifiers to the object.
Then, Tab into edit mode and select everything.
Using E
E, extrude everything ever-so-slightly along the X axis. Just barely enough to give it some thickness.
After that, add a Solidify modifier to the object, and set the Thickness to -0.0005. The negative is important, because otherwise it will thicken it the wrong direction.
Throw on a Subdivision Surface for good measure, and that should be good!
Wrapping strings around stuff is a bit complicated, but you can use the same process anywhere. Also, you can use Proportional Editing to help tweak the positions of the mesh more easily.
Hope this is what you're looking for!