Derivation of the LMMSE (Linear Minimum Mean Squared Error) Estimate and the MMSE Under Gaussian Prior
In the past I derived it as following:

It is a little different approach.
If it answers your question I will rewrite it in a proper LaTeX.
Regarding your question about the steps in the derivation you presented, it is using the Woodbury Matrix Identity (Both 12.26 to 12.27 and 12.28 to 12.29).
Related Answers:

Deriving the Matrix Inversion Lemma for RLS Equations vs the Woodbury Derivation.
Sequential Form of the Least Squares Estimator for Linear Least Squares Model.

In the answer above you may see some related derivations.