I don't think you need to stir anything. I have been adding hot gelatin mixture onto cold beer without stirring for years with overnight results. I figure as the gelatin temperature reaches equilibrium it sinks to the bottom because it is more dense and takes all the haze with it.
I agree with dmtaylor on this. Maybe you haven't picked it up because your dip tube is off the bottom of the keg.
This is my process as well. For some reason I have been getting mixed results lately. Some beers clear very fast while others will take 1-2 weeks so at that point I feel that I didn't follow the correct process or something. My current lager cleared extremely fast but that is probably also due to the nature of very flocculant lager yeast.