It's very hard to judge the damage on your chair from one small picture on my I-pad.
Steve's opinion is the chair can't be repaired. I value his opinions.
You said in your original post the chair still seems strong even with the break.
If there are any pieces missing from the break, the damage is more severe.
If the piece is broken with a clean break, I would try to glue it back because you have a matched set and you may not be able to find a replacement to match.
If you don't have time to do it right the first time, when will you have time to do it over?