-My daughter did (the) Japanese Advanced Level Examination and got into uni.
-My daughter did (the) JALE and got into uni.
-My daughter did (the) Japanese Advanced Level and got into uni.
-My daughter did (the) JAL and got into uni.
The problem with using acronyms or initials is that your audience has to know what you're talking about. Certain ones are well established, like NASA or CIA. Unless you know your audience is familiar with JALE, I'd avoid it, and write out the whole thing. Also uni seems a bit too informal when writing it out in full, so I'd say university. Finally I'd say My daughter took the...., not My daughter did the.....
It's not impossible, and I've heard people say it, but it's more correct to say take an exam. If you were applying for a job, it's best to avoid do an exam, but it wouldn't be a problem if you were chatting with friends.