[Oh Haha]

Pics are great. Wolfe.
I think gluing insulation panels on the door is your best bet. You can ocikup sheets of the stuff at most large hardware stores. This is how I would do it:

Cut the panels as close to the hinge points as possible, covering large areas in sections. I think a glue would be best and then if you can pop a couple of screws here and there it would add support. It looks like simple rectangles will do the job. You should look at the vertical seals on the outside of the door as well. They can be simple slats of wood with a rubber seal to keep the widn out. Again, your local hardware store should have all this neat stuff in stock.
Maybe the landlord would at least help with a bit of the cost seeing as how you are adding value to HIS property?