Very flexible and durable wires?
Conductive thread is readily available. There may be several materials and formulations used to make it. At least some of the available thread is very flexible, but you will need to investigate to determine how difficult it will be to deal with the higher resistivity compared to more conventional wire.

Conductive thread may not be as durable the thread used to make conventional clothing fabric. In regular fabric, quite a few threads can break before the clothing would be considered "worn out."

Making reliable connections may also be problematic. You will need to investigate solder ability or the feasibility, size, cost and reliability of other connection methods.

Your search for "LED clothing" should turn up some reports describing how others have handled the various design problems.