I'm no expert, just going on what Google can find for me.
Edit - Just looked at your photos again. Disregard this if you used all Blue LEDS in your project. I assumed you were mixing LED colours.
You said you got a batch of LED's in different colours (or colors if they are made in the US
). I think Eggscape's response is assuming all your LED's are identical. I read that different colours of LED require different current levels. I don't think they would work being strung together in series, as they all get the same current level. You probably need to wire each colour of LED in parallel with its own resistor to allow the required current level. Each colour could have multiple LED's wired in series to draw the same current.
I am assuming that the lowest current rated LED's would have blown first, and shorted as they blew. That could increase the current for the remaining LED's, causing the next bunch to blow, and so on.
I did the math on your resistor, it is rated at 0.5W. At 13V, it will generate about 0.5W with a current of 0.04A. If all the LED's shorted, and your battery is putting out a higher voltage, that could be why it fried.
I think you will need to find a calculator to help you find one to produce the correct amperage for each type of LED.
Good luck with your project, and stay safe.