Well, I think what you SHOULD do is pressure your landlord to get it sorted. However, looking at the lock there's a couple of options.
1. Get a professional to look at it (boring and expensive)
2. You can get special putty that you adding boiling water to to make it go soft and then it hardens after a couple of minutes. Given the fairly simple lock, you could do that, fill the lock with it, and when it solidifies it should form the complimentary shape, i.e. key. I think there's a couple of different products that do this, however there is a risk of breaking the lock, so, as I said, hassle landlord is the correct thing to do.