How can I make a copy of a door key without a matching blank?
If the key is for a high-security lock like Medeco or Mul-t-lock, only the locksmith who originally cut the key will be able to cut a new key as each locksmith is assigned a unique keyway.  These keys are actually not copied but cut from a specification.  Stores who cut keys but are not lock smiths might try to copy these but the results will be unsuccessful.

If the key is not a high-security key then the locksmith who said no without the original is doing his moral duty to not copy such keys since the numbering is a form of audit and access control.

The blank is not the issue - there are a ton of different types of blanks and a locksmith will know which one to use. They will not grind down other blanks normally, though I suppose if you were legitimately trying to create a copy of a very unique key (say something antique) then they might resort to other methods to copy it.

Your options are:


Find a locksmith who will copy it for you
Either replace the lock cylinder or the entire lock assembly with a new one, after which you will hold the original key and be able to make as many copies as you want. Usually you would just provide the landlord with a copy of the key so that they can enter but definitely check with your landlord to see if this is allowed. 
Have at a blank with a dremel or other type of grinding tool.  Depending on the quality of the lock, this may or may not be successful.


Of course maybe the best thing to do would just be to ask your landlord!  Not to suggest you should lie to your landlord, but if you think multiple dwellers is a sticking point then maybe just say you want a copy so someone can come clean your apartment regularly - I think you'd be hard pressed to find a landlord who would actively discourage you from keeping the apartment clean.