There's some nuanced differences between the two. I came to know implies a more lengthy, substantial or involved process of acquiring knowledge than I learned. Synonyms of come to know include come to understand and become acquainted with, which are a little more distant from learn. For example, I came to know calculus suggests a drawn out process or a deeper, more personal relationship with the subject than I learned.
I don't know of any instances where I learned cannot be validly replaced by I came to know, but I do know that the exchange doesn't always work in the other direction. I came to know my neighbor is fine, but I learned my neighbor is unsemantic (people are not facts or ideas; you cannot learn them). I discount learned a lesson here because it carries a very specific meaning and uses learn in a different sense than usual. Cambridge even lists it as a phrasal verb (or at least thinks it merits a separate entry).
However, your question includes that after both phrases. Having that makes it about some particular pieces of information. In this case, came to know is a superset of learned, because both are completed processes of acquiring knowledge, and learning is about retaining specific facts. Because of this, I can't think of any cases where one works and the other doesn't if you include that.
Regarding your example, what's the context? Was the discovery of this information expedient and easy? Is the sentence intended to be merely informational? If so, use learned. Was it very difficult or time consuming to find out the woman's status? Is it the knowledge extremely important and worth emphasizing? Are you trying to avoid being prosaic? If so, use came to know. Compare:
I flew to Bangladesh to see my cousin. After landing, I went to a restaurant to meet her and her husband. Upon arrival, I learned that she was not in town. She had to make a business trip and would return the following week.
Tracking the billionaire's wife across Europe was no easy task, but I finally found out she was in Madrid, so I double timed it over there. Upon arrival, I came to know that she was not in town. I asked around at the likely clubs and bribed clerks at the upscale hotels, but if anyone had seen her they weren't talking.
"Whether" is nearly always replaceable by "if", for embedded questions. So
I don't know whether Peter will come to school.
and
I don't know if Peter will come to school.
have exactly the same meaning: I don't know if it is true or false that Peter will come to school.
Your other sentence,
I don't know that Peter will come to school.
is grammatical, but in principle has a different meaning (though that meaning is unlikely with a first person subject).
If we switch to a third person subject:
Jane doesn't know if/whether Peter will come to school.
means that she doesn't know the truth of the question. But
Jane doesn't know that Peter will come to school.
implies that it is a fact (known to the speaker) that Peter will come to school, but that Jane doesn't know that fact.
Clearly, this distinction is unlikely for a first person subject, because the subject is the speaker. For this reason, I think, you do hear
I don't know that Peter will come to school.
with the same meaning as if/whether; but I would say this is very colloquial.
Best Answer
The English use of come and go is a little different from other languages. In English you first establish a point of view and evaluate direction from there. So, for example, if I imagine I am already at the party, I would say:
However if I am at home, I would instead say:
This can get complicated if we include another person in the sentence. In this case you can consider the point of view of that person:
or
All of this is separate from your question of inclusion or kindness. Either way can be inclusive or kind, since that depends a more on other factors like context, intonation, permission, etc.