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Reverso Context shows them both used at roughly the same frequencies for "How well", as in
"How well do you know [something or someone, like a language or subject]?"
without any subtle difference in meaning that I could see from the examples, but I could find no other info on this online, so I wanted any Russian native speakers to point out any subtle differences that mean I ought to choose one instead of the other, and which one they see more often.
if the question defines an adverb, then there is no significant difference between the two.
Как literally means "how", and насколько literally means "how much".
In English, "how much well" is not an idiomatic way of putting it, but it is in Russian and other languages.
If the question defines a verb or an adjective rather than adverb, e.g. насколько ты ему доверяешь? "how much do you trust him?" or насколько он высок "how tall is he?", then насколько is the only possible option. With verbs (but not with adjectives), it's much like in English.
