Solve for the steady state with CRS Cobb-Douglas, problem with the system of equations
1) Instead of $(9)$ use

$$C = K^{\alpha}L^{1-\alpha} \tag{9*}$$

2) Equation $(10)$ is not correct. For constant population $N$ and identical agents,

$$c^{\sigma} l^{\gamma} =  w \implies (C/N)^{\sigma} (L/N)^{\gamma} =  w \implies C^{\sigma} L^{\gamma} = N^{\sigma +\gamma}w \tag{10*}$$

Insert $(9^*)$ in $(11)$ and solve for $C$ as a function of $K$, using $(8)$ for $r$. Substitute back in $(9^*)$ and solve for $K$ as a function of $L$. Insert into $(12)$ and you will get $w$ as a function of parameters. Etc.