Let's solve this problem step by step:

1. Let the length of the rectangle be x and the width be y. Then, we have two equations: x - y = 6 and xy = 135.
2. Solving for x in the first equation, we get x = y + 6.
3. Substituting this value of x into the second equation, we get (y + 6)y = 135.
4. Expanding the left side, we get y^2 + 6y = 135.
5. Subtracting 135 from both sides, we get y^2 + 6y - 135 = 0.
6. Factoring the left side, we get (y + 15)(y - 9) = 0.
7. Solving for y, we get y = -15 or y = 9.
8. Since the width of a rectangle cannot be negative, we take y = 9.
9. Substituting this value of y into the first equation, we get x = y + 6 = 9 + 6 = 15.