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If you'd rather replace the piece it's a standard trim piece size from any big box store.
I have fixed that same area. I used Bondo wood filler, but you could use Elmers also. I taped the area off and then filled in the cracks. I let dry completely, then sanded the area down. Mine needed 2 applications. When dry, you can prime and paint.
I have and it is much easier than repairing the whole, wood, I use, glue, and paper, once it harden, i sanded and then painted.
Looks like you have maybe tried to use glue of some kind already? If that is the case, you must let it dry thoroughly and then plane it down flush with rest of door frame, before priming, undercoating and then painting.