There are some papers in which "The higher infinite II" is given as references:

1) Forcing Axioms and the Continuum Problem -Sakae Fuchino

2) The mathematical development of set theory from Cantor to Cohen-Kanamori,

3) Distributivity properties on $P_\omega(\lambda)$-Matet.

But I have no idea about if the book is going to be published or not.

Even the following content is presented for the book (note that the book is a continuation of volume I, so it starts with chapter VI)!!!!

Chapter VI. Higher Combinatorics

Kurepa's Hypothesis and Chang's Conjecture
Combinatorial Principles
Subtle Properties
The Tree Property

Chapter VII. Forcing with Strong Hypotheses I

Master Conditions (Silver's upward Easton forcing,
Kunen's saturated ideal)

Ultrafilters (structure theory, combinatorics)

Singular Cardinals Problem (intro, Solovay's
result, ultrafilters, Silver's result)

Strong vs. Supercompactness

Singular Cardinals Problem Forcing (Magidor's earlier
results)

Precipitous Ideals (combinatorics, equiconsistency
with measurable)

Chapter VIII. Covering and the Core Model

The Covering Theorem for $L$

The Core Model

Models and Mice

The Covering Theorems for $K$ and for $L[U]$

Applications of $K$

Chapter IX. Higher Combinatorics II

Reflecting stationary sets, Shelah's LM result, etc.

Chapter X. Forcing with Strong Hypotheses II

Radin forcing, Proper forcing, forcing axioms

Chapter XI. Consistency Results about AD