I have the exact same problem with my kitchen sink, no way to connect to it.
Under the sink I installed a compression 3/8 Tee to split the cold water line. One side of the Tee went back to it's normal route. The other leg of the Tee goes to a 3' SS Braided Line (Dishwasher hookup kit, $15 for the line and some misc connectors.). At the end of that line I put an inline valve, then a 3/8 barb to connect to vinyl tubing. To chill my wort I reach under the sink, grab the length of coiled line off of it's hook, pull it up, hook it up, and turn the valve.
When the wort get to 90~100, I have a small aquarium pump ($30) that sits in one of the sinks filled with icewater and I just switch over to that.
It can't push nearly as much water through the chiller as tap pressure can, but it's helps get things really cold.
$5 3/8" Compression Tee
$15 3' 3/8" Braided line
$6 3/8" compression inline shutoff valve
$6 1' 3/8" Braided line (acting as a F-F adapter past the shutoff)
$6 3/8" compression to 3/8" threaded
$7 3/8 Barb Adapter
(Yeah, once I add things up, it's more $$$ than I thought it was.. But very convenient.)
$30 Submersible Aquarium pump (Get the biggest heaviest you can afford)