Learn English – Which is best: “with”, “by”, or “using”

Looking at Google Ngrams, British English seems to use nearly exclusively "enamoured of", while American English uses both "enamored of" and "enamored with". "Enamo(u)red by" is quite rare on both sides of the pond. I would probably say "enamored of" when talking about a person, an animal, or an abstract idea, and "enamored with" when talking about a tangible object. I can't tell whether this is just me, or American usage in general. After looking at some examples on Google, I can say lots of people don't follow this rule.

He was enamored with his new model airplane.

He was enamored of the idea of running his own business.

But all three of these prepositions are acceptable grammar, and all three should be understood equally well.

Doubtless as a result of historical changes in meaning and ellipsis, the meanings of prepositions in specific modern-day expressions are legion, often etically unpredictable, and not infrequently apparently illogical. There are broad rules, but they are very broad, and exceptions are many. Avoidance of ambiguity is desirable, but not always achieved: even 'The thief opened the door with the duplicate key.' is ambiguous (instrument (= using), or identification (= which had) of the door?)

David Thatcher, in Saving our Prepositions, writes [re-formatted]:

Greenbaum (103) cites “an empty aspirin bottle was found by the deceased.” This, he says, “sounds as though the dead person found the bottle rather than, as was presumably meant, that the bottle was found beside him.”

The art section of my local newspaper ran the headline, “the world of ballet has been blessed by many fine composers,” suggesting that composers, en masse, have been usurping a priestly prerogative. By, of course, should have been with. The broad distinction is that by denotes the agent, or essential agent, of an action, and with the instrument of an action. Compare “he was struck by the sun” with “the sun struck with its rays, “the tree was shaken by the wind” with “the wind shook the tree with its strong hands, “”the city was destroyed by fire” with “he destroyed the city with fire” (examples cited by Fernald 189).

In practice, by and with are used less strictly, but “where with or at can reasonably be used instead of by, they should be” (Greenbaum 103).

That the distinction is not universally made is shown by examples such as

We must do it by long division. [internet]

and

The secret doesn't lie on whether you made it by machine or by hand but on the embroidery supplies you use to craft it. [internet]

both showing instrumentality.

But in OP's example, 'with' is the accepted choice. This is probably strongly connected with the fact that 'key' is concrete whereas 'by hand', 'by long division' ... show methods (long division being abstract, and hand in this expression intermediate).