Rick, since you've already got the cleaning kit, this may not help you, but you might want to check out the review that Bob and Gene did on the Whisper:
http://www.gatewaytoairguns.com/airguns/forums/thread-view.asp?tid=4092&mid=28077#M28077There's an place or two in there where Bob mentions using the pull-through weedeater line method (I think) and using a drinking straw in the muzzle bore area to help guide the weedeater line through the baffles without catching on them. I've read several places that it's better to introduce any cleaning apparatus from the chamber end of the barrel, rather than going in from the muzzle end where the crown of the barrel is vulnerable to damage (this goes for powder-burners as well as airguns). Also I've read here that using a metal cleaning rod and/or metal jags can easily damage the rifling in an airgun barrel due to the relatively soft metal used.
That's one reason why I have only used the weedeater line/Goo--Gone cleaning procedure, which you can find elswhere on GTA, when I first get a new airgun (haven't shot any of them enough to require a follow up cleaning, and probably won't anytime soon). Make the cleaning line from a 3 foot length of orange weedeater line and melt a little ball at the end with a lighter or soldering pencil, trimming it to fit loosely in a .177 bore, and cut a sharp point on the other end. For .177 I use a single 1" square patch and for .22 I used two 1" square patches- the sharp end of the line pierces the center of the patches. I soak the patch down with the Goo-Gone, run the sharp end of the line from the chamber end out through the muzzle end, then pull the line and patches through. I replace the dirty patches with new ones and repeat with the Goo-Gone till there is no more (or at least, very little) rust/gunk showing up on the patches. After the patches start coming through clean, I repeat the process a time or two without using Goo-Gone till the patches are completely dry. Then start shooting the hades out of it again. I haven't used any oil of any kind on the bore of any of my rifles or pistols, though others might (I do use a little on the gun's exterior metal parts for a wipe-down). I haven't seen any indication of rust throughout the humid summer.
Hope that helps.
Bryan