Remember, these are Asian Elephants. But most things apply.
How many individuals are we talking here? Elephants have too long gestation (about two years) and interestrus periods to build up a large population. I'm not an expert on Elephant behavior, but wild ones tend to avoid humans, and they don't seem to thrive well in highly populated areas (India, as highly populated as it is, still has some regions of wilderness). Most of their diet is small palms, saplings and fruit, and of course large subtropical virgin forests are hard to find in North America. And I think Elephants, as large as they are, would be quite resistant to cold; a cold spell won't kill them. The possible diseases and weakening most probably could.
The only place I can see them thriving is the American South, perhaps Lousiana and Florida, and Central America. I am not sure how the food is there, in fact they might prefer eat from introduced crops. While the idea of elephants in the Bayou is amusing, I think even those places are still too densely populated for Asian Elephants to live comfortably (African Elephants are even more migratory) and I think the local population would eventually kill them or capture them.
There is also the
US Camel Corps experiment on this period. Unfortunately, I'm not sure why the camels died off. I'm thinking they just weren't adapted to find food in the American Southwest by themselves: Bactrian Camels are domesticated after all.
They seem to be more succesful in Australia.
Interestingly, a very similar case happened in Colombia when some hippos from Pablo Escobar's private zoo escaped (a male and three females).
They now number about 50, maybe more(!), and if you know anything about hippos, they can be very dangerous to people and native plant life. Of course, such a strange case is only possible because hippos have higher reproduction rates (and they even lowered their maturity age from 3 years to 1 in Colombia), the climate in Colombia is mostly ideal for them, and they live in a relatively remote area.
Do keep in mind that Mastodons, which were basically Elephants lived in the Americas (including, ironically, Colombia) until nearly historical times, about 8000 years ago. Same with many other megafauna (giant sloths, gliptodons, etc.). There is precedent for megafauna in the Americas (in fact, some paleontologists would venture to say that our current ecosystems like the Pampas and the Great Plains are 'depleted' from major mammals). But it DOESN'T mean that Asian Elephants could easily replace Mastodons, or Hippopotamus, Toxodons. Most likely, certain particular adaptations to the American enviroments went extinct with them. And they would certainly wreck havoc in our current ecosystems.
Still, Elephants in the Pampas? Interesting thought.