It really depends on the family member and probably where they live. Here is a Southern perspective. I have heard it pronouncedmany ways. It is part of my name and I pronounce it like "angle" but with a soft "l" as the first sound. "langle"
It is NOT a hard "l" and then the word "engle." It is not "lingle" that rhymes with jingle.
It's a French name - which has been Americanized.
Luckily for me, there are lots of L'Engle experts out there who just love to tell me how to pronounce my own name!