However, it doesn't work.
In what way? What do you see? Is the LED on or off.
The circuit is right if you build exactly what you have.
Have you testes the TLC 5940 by itself in this wiring, the 1K current setting resistor is quite low make it 2K, but that in itself will not stop it working.
Always use the # icon when posting code, especially silly long winded code.
why are you mapping the values to 255?