I fully understand the concept of strobist flash as shutter. Done it myself. Cactus triggers.
So did this guy:
Strobist Golf | Flickr - Photo Sharing!
Note the problem of triggering and he *still* cannot get a crisp shot with a laser device.
The OP just wants to attempt with a K-x.
This guy did it just recently, if you can live with blur:
Golf Ball Impact | Flickr - Photo Sharing! Golf Ball Impact | Flickr - Photo Sharing!
Many amateur photographers have captured quite by accident the "wrapping" impact of a softball on a bat under natural light, usually from a distance. Or the moment after impact as the ball leaves the bat. If you slow the golf club swing down to that approximate speed and burst shoot you might get something similar with a golf club at a well-lit driving range maybe with some enhanced lighting to get the SS up. Can't avoid blur. Will take a lot of attempts. I'd use a 5-iron.
A custom trigger might work but you'd almost have to swing in the dark and have a separate trigger for the shutter from the flash due to shutter lag. If the contacts are on the ball itself and on the club head it's a race between the ball's release and the strobe. I suspect it's far more difficult than suggested because if it was easy, Flickr would be loaded with homemade attempts, and it's not. Golf club head speed and swing variability are very challenging. Video is simply how it is done now and the K-x does video and the OP has an LED light.
I don't think Bobby Jones was actually hitting a ball on that swing.