My concern would be that any wire (or whatever) that I string into the end of the snake to give it more *action* would come off and add to the clog.
After you run the snake you could try using a water jet attachment on a garden hose, it's called a Drain King.
There's at least two different sizes (maybe more?), but you'd want the small size for a kitchen drain.
It's made of rubber that expands under pressure to seal up somewhat against the sides of the drain line, so the spray has more force since it can't just drain back so easily. Run it in as far as you can before you turn on the water.
If your kitchen drain vents straight up through the roof you can run it down from there, just make sure you're past the kitchen sink entry point to the drain or it will backfill there, taking the path of least resistance. And don't fall off the roof.
It should clean up any gunk leftover from the snaking.
You could also have your drain scoped to see what's happening.