= Solution
Write dimensional variables as their scales <times> dimensionless variables, with positive scales $S_0,Q_0,m_0,N_0,t_0$. For compactness define
$$
a=\left(\frac{\sin\alpha}{f}\right)^{1/2},\qquad B=\frac{\rho_wg\sin\alpha}{L},\qquad D=\frac{\rho_iL}{S_L\sin\alpha}.
$$
A consistent explicit choice is
$$
S_0=\left(\frac{Ba}{\rho_iKD^3}\right)^{3/8},\quad Q_0=aS_0^{4/3},\quad m_0=BQ_0,\quad t_0=\frac{\rho_iS_0}{m_0},\quad N_0=DS_0.
$$
These identities give $t_0m_0/(\rho_iS_0)=1$, $Kt_0N_0^3=1$, $fQ_0^2/(\sin\alpha S_0^{8/3})=1$ and $\rho_wg\,t_0Q_0/(S_LN_0)=1$. Substitution therefore leaves the unit coefficients in all four balances, and the sole supply parameter is $\mu=M_LS_L/Q_0$. Expanding its scales yields
$$
\boxed{\mu=\frac{M_L\rho_i^2L^2\sqrt K\,f^{3/4}}{\sqrt{\rho_wgS_L}\,(\sin\alpha)^{11/4}}.}
$$
After dropping scale hats, eliminate the instantaneous quantities to obtain the useful two-dimensional system
$$
\boxed{S_t=S^{4/3}-SN^3,\qquad N_t=S^{4/3}-\mu,\qquad Q=m=S^{4/3}.}
$$
These scales assume a positive slope, positive <friction> coefficient and nonzero lake <area>; they nondimensionalize exactly the hydraulic law printed in the source.
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