***** NOT A TEACHER *****
Wow! What a wonderful question. In my 75 years of life, I had never thought about that expression until I read your post.
I found an answer in a big fat grammar book used by many teachers throughout the world. *
1. That book tells us that "Circumstantial what with refers to one or more circumstances."
a. Its example:
"What with the prices being so high, and with my wife being out of work, I can't afford a new refrigerator."
[NOTE: It is only my opinion that in "regular" English, we might say something like: "I can't afford a new
refrigerator, because the prices are so high and my wife is out of work."]
b. That book then explains:
If the sentence does not have an "and with" part, then it's OK to add "and all."
i. The book's example:"What with Carol being out of work and all, we didn't send any Christmas cards this year."
[Only my comment: I think that in everyday English, we might say something like: "We didn't send any cards this
year because of Carol being out of work and all the other [bad] things that have happened to us."]
Now let's look at your sentence:
"Something she was somewhat used to, what with being a television news reporter and all."
According to that book, we can use "and all" because your sentence does not have an "and with" section.
Now let's add an "and with" section:
"Something she was somewhat used to, what with being a television news reporter, and with her being the daughter of a famous politician, but this time it made her squirm."
Thank you so much for your question. I learned so much.
* A Comprehensive Grammar of the English Language (1985 edition) by Professors Quirk, Greenbaum, Leech, and Svartvik, page 1,106.
Student or Learner