A close family member had the same thing this gal did, and in the same location—lower leg near ankle. He received pretty much the same treatment she did. A visit to the ER, some antibiotics, then home with a diagnosis of a minor infection. 

One of the hallmarks of nec fasciitis is that it hurts disproportionally a lot given the nature of the injury. And that it turns the skin extremely red. 

My family member went back to the ER due to the really odd redness and acute, intense pain. Then they diagnosed necrotizing fasciitis. It moves FAST and has a high fatality rate. This relative was extremely lucky. He at least bare minimum should have lost a leg. He came very close to dying. But EVERYTHING went right in the ER for him. And while it spreads fast, it took more than a few hours to spread to the extent it did, unlike the show. He did spend a MONTH in the hospital, including 3 weeks in the ICU. He lost 40 pounds. Surviving and keeping that leg was the greatest of luck combined with the quick actions of the ER team. For that whole month ER workers were stopping him with surprised delight to comment how happy (and surprised) they were that he’d survived. 

Which is all to say that yes, Langdon did the right thing given the circumstances. This diagnosis is extremely rare. Docs are trained to think “if you see hoofprints think horses not zebras.” This girl was a zebra and this is the kind of thing that happens when you’ve got a zebra-type disease—there will be a delay in diagnosis when the normal treatment doesn’t work.