Transfer Function From Bode Plot
The piece that you're looking for is
$$
T(j\omega) = \frac{K}{j\omega}
$$
This has a denominator that looks like \$1 / \omega\$, so the slope of the gain is -20 dB / decade. It also has a phase of -90\$^\circ\$, which matches your phase plot at low frequencies.

You can find \$K\$ by looking at any point on the transfer function. For instance, at \$\omega = 0.01\$, you see a gain of 10 dB, so
$$
|T(j0.01)| = \frac{K}{0.01} = 100 K = 10^{10\text{ dB} / 20\text{ dB}} \approx 3.16
$$
so \$K \approx 3.16 \times 10^{-2}\$.

Obviously, your transfer function has more terms in it, but it seems like you have a good handle on those.