Yes, it had to happen...  :-D

The Kitchen Sink has been slow-draining for a while, then became Very slow-draining, and then almost completely dead.

(This is in a 2nd-Floor Flat)
I emptied out the U-bend pipe parts all the way to the wall, and it was all clear. Cleared white deposit inside the wall-pipe (limescale?), still no change.
I stupidly ran a long stick through the pipe and I think it hit a bend at about 1 1/2 feet in. It was all going fine until I was a little bit rough bringing it out and it snapped. The stick left in there is around 1-2 feet long.
I'm not sure if the bend goes left, right or down.  

So what I am desperate for help with, is:

1) How can I pull a stick back UP the pipe and out?

or 2) Would it be disastrous to push it down, with a spring or snake or something? Would that just cause a worse blockage?
I'm thinking of some kind of grabber, like a smaller version of one you pick up rubbish with, to perhaps grab the stick and pull back. (ie- http://www.instructables.com/id/Green-Garbage-Grabber%2c-Trash-Tongs%2c-pick-up-... ) Does such a tool exist?

Any help is massively appreciated!!