What is the role of the な in bold in this sentence? Is it a nominalized だ?
考えれば考える程彼女がどういう女なのか分からなくなってきます。
Let's start with this part of the sentence:
彼女がどういう女だ
We'd like to add のだ, which includes the nominalizer の. When we do this, だ changes to な, as you suggest:
〜だ+のだ = 〜なのだ
Here's what our sentence looks like so far:
彼女がどういう女なのだ
Since this has a question word, it's a question clause even without か. But we can add か anyway, and in this case it's required because of the following verb:
彼女がどういう女なのだか・・・
If this were a complete sentence, we'd want to delete だ before か:
彼女がどういう女なのか ← か, not だか
But since the question clause is used as a complement for the following verb, deleting だ is optional:
考えれば考える程 [ 彼女がどういう女なのだか ] 分からなくなってきます。
考えれば考える程 [ 彼女がどういう女なのか ] 分からなくなってきます。
The last one is your sentence, so hopefully you can see how it's put together now.