From the pictures it looks as if the head stock is a mortice and tenon type. I've never seen one of those before. It appears as if the guitar might have fallen over, like mine did, and the glue failed. I'm unsure of the make but I can say it is not a Kay or a Silvertone but may be a Japanese guitar from the 1960's. I take it there is no label inside the sound holes?
If it were me, I would steam off the original veneer. Use a steam iron with lots of steam. Lay an old T-shirt/ cloth on top of the veneer. After a few minutes, try to separate the veneer with a razor blade, keep applying steam and heat until you can work off the old veneer with razor blade and putty knives. This step is just for cosmetic reasons and not necessary to repair the guitar. Some people would want to leave it as original as possible.
Apply plenty of glue to both the headstock and the neck, work it into voids with a paint brush or tooth pick. clamp the two parts together the best you can. Wipe off glue that oozes out with a damp rag. Let it sit for at least 24 hours.
While you're waiting for the glue to dry, start shopping for a new veneer for the headstock. There are several luthier shops on line that can provide any kind of wood you can imagine. Good luck, looks like a fun project. Keep us posted!
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