Find the value of k if $$13 - \left[10 - \left\{16 + 8 \div \left(k - 2\right)\right\} + 2\right] = 20$$.

A

$$\frac{14}{3}$$

B

$$\frac{31}{3}$$

C

$$\frac{40}{13}$$

D

$$\frac{23}{13}$$