Disclaimer: I don't even have an e-cig yet, so I'm just going off the stuff I've read and seen.
Considering some of the procedures I've read here, this doesn't seem too bad and just might work:
1. Soak in baking soda water. Use warm to hot water with as much baking soda as will dissolve, soak however long you like, 5 minutes at least as a guess. Baking soda will be basic and may dissolve some of the components of the gunk. Like many methods, some people say this has worked for them by itself.
2. Soak in vinegar. Or any other acidic solution, like lemon juice. On the one hand, the acid may dissolve some other components of the gunk, on the other, reaction with the carbonate from the baking soda makes carbon dioxide (it fizzes). Since the baking soda soaked in, some of that bubble production should happen down inside the wick and gunk, and might help break it up. Since there's no real way to scrub the atomizer, it seems like it couldn't hurt. May be part of why cola has worked for some people, carbon dioxide bubbles forming in the wicking, (plus cola is acidic). Soak until fizzing stops, and then a bit longer just to be sure.
3. Rinse. I'd suggest filtered or distilled water, but it probably doesn't matter.
4. Dry.
5. Drip, gotta re-wet all that wick with juice.
6. Test?
As the disclaimer says, I have no way to try this, so if anyone does before I get a gummed up atomizer to play with, lemme know how it turns out.