Probably units and the format of eq (7) are the problem. After looking at the referenced website by Nakka, I used the questions information given to get $\dot m= 1.187 kg/s$ (using a rounded off r=12mm/s= .012m/s). Also calculated throat pressure and temperature of 3421000 Pa and 1616 K. Equation (7) should have parentheses around the $ w_t/P_t$,because the pressure and temperature are divided. Eq (7) $w_t$ is $\dot m$. The $g_c =1 (no units)$ when using SI units. The $R = 8314/42.39 = 196.1 m^2/s^2 K$ (units simplified from J/kg K).
The $\dot m/P_t $ has units of (kg/s)/Pa, but Pa (pascal) is N/m^2 and N is kg m/s^2 so m^2 s/ m. The square root of RT has units of m/s, so the result is $m^2$.
Using values above I got 1.837 10^-4 $m^2$ or 183.7 $mm^2$
The mass burning rate = $\rho r A$ where $\rho$ is the solid propellant density, $r$ is the burning rate (how fast the solid is consumed) and $A$ is the burning surface area.
As given above, $r=a P_c^n$, for design chamber pressure of 5.86 MPa, a and n are 3.84 and .688 so $r=3.84 (5.68^.688)=12.69 mm/s$. A trick here is to realize the pressure is in MPa. Using the english units, the rate is near the same (850 psi): $.005(850^.688)=.5181 in/s=13.16 mm/s$/
The mass burning rate goes out the nozzle, so calculate $\rho r A$ in SI units: $\rho = 1.785 g/cm^3 = 1785 kg/m^3, r= .012 m/s,A=55411mm^2=0.055411 m^2$, mass flow = 1.187 kg/s. This mass flow goes to the nozzle so it is used in eq (7).
The units on $\sqrt( RT) $ are m/s. If we multiply eq (7) by $P_t$ the left side is a force (pressure times area), the right side has mass flow rate times velocity and that is a force so the units are correct. Remember the basic rocket thrust force equation is mass flow rate time exhaust velocity.
Just for completeness, Eqs (7), (8) and (11) can be combined to show flow rate, chamber pressure and temperature together. From Rocket Propulsion Elements by Sutton (7th ed, but 1st edition has the same with same eq number):
Sutton uses k for $\gamma$,other symbols you should be able to figure out.