Daisy-chaining 7 IC's

Sounds like its time to firm up the assembly.
Get some perfboard, some wirewrap sockets, and wire up a real assembly.
Wirewrapping is nice, goes quick, easy to correct if you miswire something or need to make a change.
This board only took an evening to build up for example.
You can see the '328 & crystal/caps that make up an arduino, and some shiift registers above it.
On the left is a MIKROE483 FTDI module for USB interface to a PC.
Add your TLC5940s and LEDs next to them, be an easy wiring job.