How do I normalize a predesigned IIR filter?
As mentioned in the comments, you have to choose a frequency $\omega_0$ at which you want to normalize the gain. This could be DC (i.e., $\omega_0=0$) or any other frequency, depending on the filter's characteristic.

You have two filters in series, each with transfer function

$$H_i(z)=\frac{1}{1-a_iz^{-1}+b_iz^{-2}},\qquad i=1,2\tag{1}$$

The total transfer function is just the product of the two individual transfer functions:

$$H(z)=H_1(z)\cdot H_2(z)\tag{2}$$

The gain at frequency $\omega_0$ equals the magnitude of the total transfer function $(2)$ evaluated at $z=e^{j\omega_0}$. So if you scale one of the two transfer functions $H_i(z)$ by the inverse of that gain, your total system will have unity gain at $\omega_0$.

Scaling in your time domain equations simply means scaling the input variable.

Note that $\omega_0$ is a normalized frequency in radians:

$$\omega_0=2\pi\frac{f_0}{f_s}\tag{3}$$

where $f_s$ is the sampling frequency.