Hey there,
I agree with the others; cut off the healthy part and it should work. Here's a method for rooting cuttings that works for me:
I root all of my Nep cuttings in a terrarium then slowly harden them off once they've rooted.
I generally use a 10 gallon aquarium, but if I have a lot of cuttings (the last time I took cuttings, I had 16) I use a 20 gallon long. The "long" aquariums have a larger footprint which gives me more room for cuttings and they're also a lot shorter than a regular 20 gallon aquarium which gets the cuttings closer to the lights. I put about an inch or so of water in the bottom of the tank, and put in some eggcrate which I keep out of the water with little pieces of PVC pipe. I keep the tank almost completely covered, except for a little space that runs along the entire back of the tank. I put four T12 flourescents across the top of the tank to make sure the plants have plenty of light.
I make the cutting at an angle to make sure there is as much surface area as possible at the base of the cutting. I make almost all of my cuttings with one node and I dip the cut end in rooting hormone. I wrap that in wet LFS, put it in a little 2 inch pot and stick it in the tank. I sometimes cut the leaves of the cutting in half. By doing this, you cut down on transpiration and water loss from the cutting, but if you've got the cutting in a really humid environment, I don't think this step is necessary.
You want to keep the LFS around the cutting moist, but not WET. I've read where keeping a Nepenthes cutting too wet will hinder it producing roots, but if it's kept a tad drier, the cutting will put out a much better root system. I haven't tried that yet, but I've got some N. x ventratas that I'm willing to see if it makes a difference.
Using this method, I've had cuttings that started pitchering before they were potted up. As a side note I only make one cutting nodes because I've made cuttings with two nodes and both of the dormant buds sprouted and I'd have to let them both grow for a while so they'd be big enough to make another cutting so the plant would only have one growth point. I'm anal retentive like that sometimes.