Why not i sin ryg?
I wonder that also. Google translate seems to agree.
Google Translate is by no means reliable though. In Danish, it is usual to take a simple definite instead of a possessive pronoun when talking about someone's body parts. Therefore, 'ryggen' is preferable over 'sin ryg' here.
The "her" in "her back" needs no equivalent in Danish?
It's fairly common to use the definite suffix when it's more obvious who the thing belongs to in Danish