Ledge around staircase landing instead of handrail

I currently rent a 2 story single family home. Although I am not the homeowner I'm hoping this post is still allowed. The second story of the house is an attic conversion and there's a ledge around the opening of the stairs that's about a foot wide and only two feet tall so definitely not up to code. There is a handrail along the wall that's the appropriate height. I have a little girl and I'm afraid that as she gets more mobile she could get herself onto the ledge and then fall down to the first floor. Right now I just keep her away from the second floor almost entirely.

I'm planning on asking the landlord to look into bringing this up to code. When I do so I would like to suggest some options to help move things along. Would building the ledge up bring it into compliance? As a (probably) cheaper option would it be okay to install a railing that sits on top of the ledge or beside it?