How would one extinguish a growing kerosene fire (say, when a hurricane lamp is spilled on a carpet)? You'd do better with a more solid floor. Let's say you have 2 cups of kerosene in the lamp and the lamp is dropped. Two cups in a small area (such as soaking into a rug) isn't going to provide much of a fire. What you're looking for is a splatter effect. The lamp smashes against a solid surface after a fall of three or four feet. Glass and kerosene are splattered around in a decent size circle, with the kerosene igniting because the lamp is lit. Depending on the splatter, you could end up with an area of fire that encompasses several square feet.
What is the effect of water?--I'm hoping it doesn't work very well b/c my character is about to try it. Water will just spread the fire, unless you have a significant volume. Twenty gallons might put out 2 cups, but I wouldn't want to try it with anything less. In other words, you stand a chance with a garden hose, but throwing a bucket of water will do nothing for you.
Then he's going to try and stomp it with a twisted ankle, catching his pant leg--ouch! He's probably not. It's going to be too hot to want to try this. However, definitely a splatter could end up on his leg and start burning his pants. This is one of the big dangers with flammable liquids.
When and if neither of these things work, what household item would do the trick? Assuming he doesn't have an ABC extinguisher, a dampened towel or blanket would be good. Baking soda is very effective for fires but probably won't have enough to work. A potted plant sounds interesting and would be effective. Inflammable seat cushions can be effective (use to smother).
Remember that whacking on flammable liquids usually just causes them to spread.