Final answer:
The gluteus maximus and tensor fasciae latae are the muscles attached to the iliotibial band, which stabilizes the knee and assists in thigh movement.
Step-by-step explanation:
The muscles that are attached to the deep and superficial layers of the iliotibial band are gluteus maximus and tensor fasciae latae. The gluteus maximus is part of the gluteal muscles which also includes the gluteus medius and gluteus minimus known for extending and abducting the femur. Meanwhile, the tensor fasciae latae assists in flexing and abducting the thigh, and works with the iliotibial band to stabilize the knee. Therefore, the correct answer to which muscle is attached to the deep and superficial layers of the iliotibial band is both a) Gluteus maximus and b) Tensor fasciae latae.