Lights out!

Mmm, well, I don't have the piece, but resistors are usually pretty durable, so I doubt that's the problem; if you can get to the wires at both ends of the resistor, a simple multimeter check should only take a minute and resolve the possibility.

As to the LEDs, are they wired in parallel, or serial? If serial, yes, one blown LED *would* take out the whole string, but if parallel, the whole string should light up with a battery applied to any of them (other than any blown LED). If you can, arrange a set of 'jumper cables' from an AA or AAA battery to each LED in turn (most individual LEDs will handle up to 3 volts, so the one and one-half volts from the battery should be safe enough for a test; if the individual LED polarity is unclear, try connecting first one way, and then the other, until the LED lights), after first unplugging the string from the board.

FWIW, I suspect you may just have a bad solder junction (notwithstanding the visual inspection), which a continuity test across the plug connectors may confirm (try the probes both ways, and if both orientations show an open circuit, move around the circuit testing each junction for continuity), *or* there may be something wrong with the supply voltage from the board (a voltage check across the board's plug terminals with power on should confirm or eliminate this as a possibility). If either suspicion pans out, IMHO it might be easier to just return the piece than try to repair it. Good luck, in any event!

No problem. (You *did* try them with the polarity reversed as well, right? All LEDs (being diodes, after all) are absolutely polarity sensitive, and will work only one way, even if *not* defective. Also, did you perform all the other tests I suggested?)

As to next steps, are the defunct LEDs wired in series, or are they in parallel? I doubt you will find a "string" of LEDs that will be a plug-and-play fit for either, but I think if you pick up a bunch of individual LEDs of the right size and color, and connect them with your own wires and salvaged (if still good) or replacement current-limiting resistor, you can double check whether the built-in power supply is still good (you tested the output, right?) and confirm that replacing the LEDs will in fact solve the problem. If the power supply proves it can power the replacement LEDs, you can solder on the existing connector from the LEDs to the board onto the LED string, so you can plug the new "string" into the (tested) power source, after you fit the LEDs into the structure.

It's eminently possible to wire your own LED projects, with basic soldering skills. For instance, I built a "renewable energy" billboard from scratch, wiring 18 color changing and flashing LEDs in parallel to run off a single 3.2 volt rechargeable battery, wiring each individual LED to the two bus bars (LEDs and bus wires are hidden behind the cover):



Here's the billboard on the layout, being activated (along with the 'windmill') by a button under the edge (note: the billboard colors/flashes are less visible due to the viewing angle):

Thanks, but no great skill involved -- just be sure to pay attention to polarity on the LEDs, protect them from excess soldering heat by using some form of clip-on heat sinks on the leads (I just use regular alligator clips), and be sure you wire them in the same pattern as the original. Also, be sure you verify the status of the power source first -- no form of replacement LEDs will do you any good if the board powering them is defective! After you get your replacement string working with the power supply, you can begin the tedious task of removing the old LEDs and fitting their replacements! 

In any event, good luck!