1 - It cant be repaired.
The only way to repair it would be to CUT it in small pieces and make new "replacements".
Oh, this is too pessimistic :: it should be quite possible
for a wood turner to simulate the missing part of the
broken side pillar to be fit up into the chair back-top
and then, with matching angle-cut in the new & old
pieces, to bond the old w/new!
2 - It has sentimental value, so i wont buy a new one.
Good for you! (Too much is thrown out in USA --WAY too much.)
It might seem that somehow a good deal of supportive pillar
is missing on this broken flank, but I believe that as far at
most forces are concerned, this wood here served more as
a "tension member", not compression.
It is likely --given the cordage you are showing,
which has some visual blend w/the wood--
that you don't need any special >>knot<<,
but just need to press the back into the state
you want to hold it (the getting-other-pieces-IN)
and then make several lashing wraps from under
the seat around the near chair leg and up --not
quite the same angle as had come from the
broken piece, but close enough?!
And then it's a simple matter to tie off the end(s).
(One might try putting in more tension by then
"frapping" --in a sense-- these lashings by pulling
them w/separate cordage towards the base of
the broken piece; but one cannot have much
lifting force on that.)
--dl*
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