I've seen repairs for this kind of failure. You are in "fun" territory
Rebuilding the bent frame of the back is unlikely an easy job.
As the rest of the back is still in place, you'll want to rig up a way to work on the chair upside down with the back clamped as well as can be in place.
You can drill into the bottom of the broken back. Maybe 2 or 3 inches if you can. I'd not suggest more than 1/2 the diameter of the back.
Then insert a dowel that can be used to fill the hole. Likely need to create something stepped. So you can fit the top half in the back, but fill the hole in the seat. Here is a good place for epoxy. It can not only add a bit of extra strength, but you won't need as precise a fit. Might research how much flexibility you'd want in an epoxy for this joint.
Don't use a dowel from the home center. Either make one or seek out one that is as much straight grain as possible. I'd also look for oak or maybe ash. Something that can hold up to some racking.
If you had access to a major hollow chisel mortiser, you could make a square hole that would have much better glue surface and strength. But thats a pretty big machine to work around the chair.
Sadly, there is no way the chair will be as strong as it was.
Another more radical option is to make a longer cut in the back and sister in a new piece. That would be the strongest fix, but may require extra effort to match the stain and be a more involved in place cut.
Best of luck!
Oh, hi to Katy!
Jim