I would use one of those peal off paint safe picture hangers to pull pull the broken bits back together after I put a bit of wood glue on them.
Daughter needs to help with this repair, IMO.
I put 2” wide painters’ tape and burnished the edges, before putting a large thin skim coat of drywall mud over it.
Then sanded when dry and repainted the whole door.
I would put wood filler and tape it and then paint it a bit.It doesn't seem that hole is big