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Actually, I misspoke, I never use starsan (nothing against it) it is b-brite that I use as a soap.
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The next time you are cleaning, disconnect the line, and attach to to your auto siphon.
Use the siphon to push a bucket of soapy water through the line. Then push some clean water through the line, then (because I'm paranoid) push an iodine solution through the line. To dry, attach it to your CO2 and let the tank pressure push most of the water out.
If you time the cleaning of your line with other cleaning efforts, you don't even need to mix solutions. Often I'll use the same soap (starsan for me) to do my keg, carboy, and lines at the same time.
One thing to keep in mind is that most lines are very inexpensive. About 1/3 of the time, if my lines look nasty, I just undo the clamps and replace it. Use properly sized automotive clamps, so a quick couple of turns with a screwdriver will have things apart before you know it.