How did you get \(I_R = I_L\) from \(I = I_R + I_L\)? You need to define \(I_R\) and \(I_L\) in terms of \(R\) using Ohm's law, \(V = I \times R\). Just pretend the bottom terminal is ground (0V).
With the top node as A and bottom node a B, I = I_R + I_L simplifies to:
I = (V_A - V_B) / (2R) + (V_A - V_B) / (R)
= (V_A- V_B + 2V_A - 2V_B) / (2R)
= 3V_A - 3V_B / (2R)
\(I_R = \frac{V_A-V_B}{2R}\\
I_L = \frac{V_A-V_B}{R}\\
\frac{I_L}{I_R} = \frac{\left(V_A-V_B\right)/R}{\left(V_A-V_B\right)/2R}
= \frac{2R}{R} = 2
\)
\(I_L\) is 2x larger than \(I_R\).
Another way to look at this is that current is inversely proportional to resistance, or \(I \propto \frac{1}{R}\). This gives:
\(I_L \propto 1/R \\
I_R \propto 1/2R \\
\frac{I_L}{I_R} \propto 2R/R = 2
\)