I’m not immediately sure what the problem might be, but it certainly isn’t R10; that resistor is only connected to the red ring of LEDs. ;)
* The blue LEDs are controlled through R1 – R6. So, it would take several independent problems– very, very unlikely –at either the resistors or LEDs to cause all of the blue LEDs to not light.
* It is otherwise conceivable that there could be a power supply problem or major short circuit, but the fact that you have 5 V on pin 1 seems to rule that out.
* It is possible that the LEDs are just turned off in the normal “sleep” mode. Try pressing the three buttons on the bottom, one at a time, to see if it the LEDs spring to life.
* It is possible that the microcontroller is not operating because of a problem with the crystal oscillator. Carefully check the solder joints at XTL, C1, and C2. Make sure that C1 and C2 are the correct capacitors, which should have come with a black stripe in the kit.
* It is possible that the microcontroller is not operating because of a problem with the power supply connections to the chip. Check carefully the pins of the chip, to make sure that there are not any “dry,” cracked or missing solder joints there.
* If none of those things helps, it is (unlikely but) possible that there is a problem with the firmware on the clock. If you have access to an AVR ISP programmer, I can help point you in the right direction, or you may want to consider sending it back to us for repair.