Without more context it's impossible to say exactly what nuance is intended in OP's example.
To get to [do something] can mean have the chance to do it.
To get to know someone/something can mean become [very well] acquainted with them/it.
So OP might mean he's glad he came to know her [well], or glad he had the chance to know her.
You might think it's stylistically awkward to use both senses consecutively, but there's nothing grammatically incorrect about...
The NAFEC duty was boring most of the time, but gave me a good chance to get to get acquainted with the crews.
There are many other meanings for to get, including have got to = must, should, are obliged to...
"I've got to get to know her"" (I'm compelled, or I really want to get to know her).
...and to get = become, arrive at a point where, reach a state of...
...he saw more and more of her every day, until he got to be restless and nervous when he was not with her.