Draw a plane which intersects the open end and then trim it with the chair. Next, join the newly trimmed piece at the open end to create a flat end. Join all surfaces. Divide chair in half, delete one half and then mirror the newly capped piece.
The problem is the end is capped flat, which is not what your contours show. Once you have a solid chair with the flat caps, you can then start to boolean cutaway the end parts to sculpt the capped sides so they are not flat.
IMO, it's a difficult seat for a number of reasons. Once you cap, you can't fillet easily because you have too tight corners. You could also loft pieces in place using the existing end profile, and make sure your last loft piece is planar so you can make a flat surface to construct your solid.
If it were I, I'd try and rebuild the chair using lofts. Or else consider using SubD modeling.
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