Density of integers with many prime factors

This is answered in:

Mehrdad, Behzad; Zhu, Lingjiong, Moderate and large deviations for the Erdős-Kac theorem, ZBL06553541.

(can be found on arxiv.org) The paper also has an excellent bibliography, with many related results cited.

As a survey in book form I would recommend Tenenbaum's book (Introduction to analytic and probabilistic number theory), chapter II. 6.1 (Integers having $k$ prime factors). Also the notes of the end of the chapter give very useful references (such as the Hildebrand-Tenenbaum paper mentioned by Lucia, the Selberg-Delange method etc.). I would doubt that extending the large deviation techniques from $\omega(n)$ about $\log \log n$ to say $(\log \log n)^2$ is of great use. These end of chapter notes rather direct to Hildebrand-Tenenbaum.