He never told her his name was Kal-El and she always called him Superman. Even Clark didn't know his Kryptonian name was Kal-El until "The Foundling." This was the episode where
Lex discovered a lot about Superman, including his real name (I think).
So, no, we never saw her find out about his birth name. Or, if she did, it was so uneventful and unimportant that I've forgotten about it

.