I have a super easy, no faucet or pump solution that works great. I attach my immersion chiller to my bottling bucket's spigot with regular 3/8 inch plastic hose, and another 3/8 inch hose that I send to the sink. I use the Screw-Type Metal Hose Clamps to attach the hoses to the chiller.
Right before I begin brewing, I fill my bottling bucket with as much ice as I can find. I use a combination of blocks and cubes, about 15-20 pounds worth. After I add the ice, I put the bucket up high - on top of the fridge works great - to create enough water pressure to force the water through the cooler. Then, I top off the bucket with cold tap water. I find that an empty plastic water jug works great for this.
When it's time to cool the wort, just double check the connections and open the spigot. I would keep the kettle on the floor, so that you have maxim height difference between the bottling bucket and chiller.You'll be sending a wonderfully cold 40 degree water through your chiller. The key is to add more water to the bucket as the level starts to drop.
Using this method, I dropped my 200+ degree 5 gallon wort to 75 in 20 minutes. I did gently stir the wort every couple of minutes, and I use a 25 foot chiller. I went though about 10 gallons of water.
All you need to buy is some additional 3/8 hose. I wouldn't reuse the hose used for chilling for racking purposes as the ends do get a bit hot from the copper, and that slightly distorts them.