I didn't have enough time yesterday to type up a proper answer, but I wanted you to have at least some information. For completeness, first my initial response.
Either the breaker has failed or there's a bad connection somewhere along the line. Check for voltage at the breaker and in each box along the line, the last box with voltage has a bad connection, probably one those push in connectors.
Based on your response, it sounds like the problem could be in one of the porch light connections where the power should continue on to the kitchen lighting. Which leads into your point to point question.
The wiring does loop back to the breaker, but not in the way you think, unless you have war time British ring wiring. Other than ring wiring, all conventional wiring is either linear or tree organized with pairs of wires (plus ground hopefully) making each run between boxes. Current runs up one wire, usually black in the US and brown elsewhere, and back along the neutral, usually white in US and blue elsewhere. In other words, the out and back is achieved in the same cable with a wire pair, not by separate cables. Assuming conventional non-metallic (romex) cabling.
Actually, alternating current doesn't go anywhere, it vibrates back and forth, but the direct current model serves conceptually to ensure power and neutral connections are properly maintained. Since circuit layouts can be almost anything, without a wiring diagram, only methodical checking of various wires in each box will reveal the actual layout. It's usually a safe assumption that the layout is roughly efficient in wire use, you're unlikely to see runs double back on itself without good reason.
Since the porch lights work on the same breaker and are fine, yes the breaker itself can be eliminated as a problem. It is possible the power connection that has failed is located in the breaker panel though, as two separate lines out of the same breaker could be possible, though somewhat unusual.
With that background, you should try to locate where the power runs from the porch lights to the kitchen lights. It may be from the switch or the light, or it may be a branch farther back towards the breaker. There may also be another box for something else between the porch and kitchen lights. You will need to guess at the probable layout to have any hope of locating the bad connection without testing the entire circuit. Wires typically run from closest box to closest box. Don't forget switches in two different rooms can be only inches apart for wiring, the boxes only happen to open into different rooms.
Also verify the power comes from elsewhere to the switch and not to the light. If there are only two wires, one cable at the switch, the power is coming through the light first, where you will find at least two cables and four wires. Unfortunately, there are also layouts involving three wire (plus ground) cables where where the power comes from is ambiguous.
A lot of talk for all you really are trying to do is find the other end of the wire which should have voltage but doesn't. Unfortunately, finding it is either a lucky guess or being thorough and methodical. I wish I could help more, but I can't be there to help you, but I am here fairly regularly if not frequently if you have any more questions. Good luck.
The comments pretty much answer it, but the important take away here is that the wire should possibly be replaced. I was a little thrown by the "speaker wire" comment but I presume you are referring to the light.
Get an electrician in there with a MEGGER to test the wiring. They will determine if it is compromised in areas unseen, and recommend replacing the run if needed.
Be safe! Get the megger. Chances are it melted where you found it, but it could be bad elsewhere, that you haven't found it.
Best Answer
Does it really need (not should) to be replaced: Is there a lifetime failure count for a breaker? -StackEx
Here's some extra Yahoo answers nonsense, from my attempted search in finding a way to safely make a breaker pop.
The other answers to your question here bring valid concern about replacing a questionable breaker. I'd be more concerned with the condition of the wire, which I believe was the noise (rattling in the pipe). Circuit breakers are pretty good about failing in a safe state, I.e, it just won't work anymore. I wouldn't be so concerned that a monetary/slight (crappy) contact didn't trip the breaker; you did not achieve a dead enough short (see wallyk's). It was not a good thing to have happened (although not the worst) and if you're not going to pull the wire it may be safer to replace the breaker with an arc fault.
THQL1120AF, an arc fault for $50, except it's not a tandem. No good unless it happens to be the one single you have there. aplussupply.com sells standard replacement breakers for your panel. I was unable to find a THQL1120 (AF or GFCI) tandem breaker. FYI, I looked up a 100amp AF for that panel, it's $800...
You could swap that single for a new regular tandem, freeing an entire slot (by taking the other wire from the bad breaker (provided that it was on the same leg originally) for a 1" AF, now hooked to the circuits' questionable wire. Request clarification if you do not understand the importance on which of the legs a breaker is on and how to properly relocate them.
Go big or go home. Meaning, as discussed here, replace that panel or leave it alone as much as possible.