I know that this won't totally answer your question because we use different methods of chilling the wort.
To me it depends on whether I am using just pellets or a combination of pellets and whole cones. If I am just using pellets, I use a hop bag so I can minimize the possibility of clogging my inline screen and plate chiller and pull the bag at flame-out. Since I use a false bottom in the kettle and draw wort to be chilled from the center of the kettle, I can't whirlpool. Someday I will get off my lazy ass and change the pickup in the kettle so I can whirlpool and eliminate the bag. I will then leave the pellets in the kettle until all the wort is drawn off. Plus, this way I can drain some of the wort out of the hop bag before cooling.
If I am using both cones and pellets, I just leave them in. the kettle because the cones will act as a filter bed for the pellet residue (as mentioned by ynotbrusum). Remember leaving hops in the kettle will increase the bitterness a bit but it should not be that noticeable to most beer drinkers. If you are a purist, you may want to compensate for the extra bitterness you will get by leaving the hops in the kettle a bit longer during the chilling process (brewing software will show you how much extra bitterness you will get with this method, or you can calculate it manually), but I just tend to ignore the difference.
Here is another thought. To make it easier to whirlpool, you might want to pull the immersion chiller out of the kettle, whirlpool and stick it back in, or put the immersion chiller in the kettle immediately after whirlpooling. The wort will still be hot enough to kill any nasties that might cling to the chiller.