"she can keep on drinking" is fine but the "on" is not needed and it should be counted if you don't have it in your translation.
Did anyone else hear "viviendo"? This is the first time I got caught with the b/v sound issue.
Yep got caught - I think it would be much easier to understand if it wasn't a computer generated voice.
I agree with Acott, "on" is not needed! have a look: http://www.collinsdictionary.com/dictionary/english-french/to-keep-doing-sth
How does this sentence differ in meaning from "Ella puede continuar beber"?
The idea is that to keep on = continuar. To keep is not always the same as continuar. In fact if you say: keep the keys, you can translate it as: "guarda las llaves" o "ten las llaves". This is why they offer continue in the translation.
It is not always the same, but in this case it is. Just because "keep" has other meanings in other contexts doesn't mean that it isn't appropriate in this one.
Could you also say, "Ella puede continuar a beber?" I know that isn't a gerund, but would it convey the same message?
No it does not work. It would be "understandable" (I have seen this mistake from certain italians), but it is definitely incorrect.
As an English speaker first, I agree with Acott and FraAriel- "on" is not necessary.