I put her name in the search and at the top of the results list was her record in the 1930 Census. Here's her whole family. You'll notice that her mom was also named Gertrude:
The second result was her Footnote Page created from the Social Security Death Index, to which I added some photos, a fact and a quote, mentioned in the NPR story, from New York Mayor Jimmy Walker who told Ederle, "When history records the greatest crossings, they will speak of Moses crossing the Red Sea, Caesar the Rubicon and Washington the Delaware, and, frankly, your crossing of the English Channel will take place alongside these."
Then there were several newspaper articles about her including these two from before her swim (Footnote only has the San Francisco Chronicle and Chicago Tribune through 1923):
And this one from The Roundup Record of Musselshell County, Montana defending her swim against criticism from "various English newspapers"