Even though I am a native English speaker, I am not quite sure how to interpret the English translation. Because the present perfect form of the verb is used it must mean I have just written the letter. Because of this confusion "now" is rarely used with the present perfect. It is more commonly used in present progressive sentences like this: "Now I am writing the letter." Please advise.
I am not happy with how participles are taught in English language courses for Esperanto ... including this one. It's best to think of them as adjectives with inflexible meanings. "La kuranta viro" is "the running man". "La skribinta persono" is "the person who has written."
It's possible, then, to take these adjectives and use them with "estas". So, just as a "alta viro" can say "Mi estas alta", a "skribinta persono" can say "mi estas skribinta" or even "mi estas skribanta leteron".
The trouble comes when we try to think of "estas skribanta" as a kind of verb, and not as an adjective. In fact, the best translation for the English phrase in the OP is "mi skribis la leteron." If you need to clarify WHEN the letter was (or had been) written, this should be done with additional words - like "mi jam skribis la leteron kiam li alvenis" - I had already written the letter when he arrived.
try "now I have written..." and it should make more sense to an english speaker. Other languages use "to be" instead of "to have"