Perhaps I can offer a better solution? I made my immersion chiller like you plan to do, but instead of using (and wasting) loads of water from the faucet (honestly, the water coming out of that thing isn't very cool in the first place and is an awful waste of water since the exhaust water goes down the drain). I used a pair of vinyl hoses, the 'exhaust' goes into a mop bucket full of ice, the 'intake' is attached to a cheap pond-pump (from Lowes, but any hardware store and probably walmarts' fish tank area should have them) -the pump is submerged in the ice water, this recirculates the water and melts the ice as it goes along. It cools the wort much faster than the faucet water does, and doesn't waste gallons of water.
Now, for your particular problem, however, one thing that MIGHT work (depends on how large the outlet is on your faucet) you might try looking for a drain adapter for dishwashers (I had to get one of these for my dishwasher, it was at the local Lowes store) -this thing is a rubber coupler that is just soft enough to slip over the outlet nozzle of the water pump on the dishwaser, the other end steps down to be clamped inside the drain hose that leads to the drain under the sink. It MAY be large and soft enough to fit over your sink faucet, if it does, it should allow you to connect it (If you don't want to cut off the coupler on your chiller, then use an adapter to connect the chiller to a vinyl hose, and slide that over the rubber adapter I'm talking about).
Hope this helps.