Congratulations! 

you don't like the traveling salesman solution (J I H G K F A B C D E) ? ;-D

I'll give it another try tomorrow. In case nobody comes up with a neat solution I'll write an email to Nathan Haselbauer (the guy who made the quiz). 

I guess the traveling salesman solution is probably the most palatable of the answers I've seen. What I don't like about each of the solutions people have proposed is that they don't use a lot of the information in the puzzle, like why are there bulbs on some of the connector ends and not others? Why are there multiple lines connecting some letters and not others? And what is the significance of the inner circles? In my experience, it's very rare for IQ tests to include useless information in puzzles (which is of course the opposite of real life), but maybe this is a counterexample and the extra stuff flusters dumb people more than it does smart ones.

I'm still guessing that this has no right answer and is put in the test to throw people off, although I realize this is an extremely vain attitude: "if I can't solve it, there must be no solution"! 

I PMed Nathan H. yesterday; here is his reply:

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Unfortunately I don't have the explanation to this one. It was designed six years ago by Mike Dickeheiser, and he's no longer active so I don't know how he came up with the answer.

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Maybe Dickeheiser spends all his time trying to solve his own puzzle ;-D. Kidding apart, I think the puzzle has a non-elegant solution with a touch of arbitrariness.