Well I'm 3 years late to this party but thought I'd add this as the simplest method I've found for arriving at 587 BCE using only WT literature.
Zechariah 1:1 NWT
"In the eighth month in the second year of Da·riʹus, the word of Jehovah came to the prophet Zech·a·riʹah son of Ber·e·chiʹah son of Idʹdo, saying: ..."
it-2 p. 1225 Zechariah, Book of
"Since it was in the “eighth month in the second year of Darius” (October/November 520 B.C.E.) that “the word of Jehovah occurred to Zechariah” (1:1), the book covers a period of at least two years."
Zech 1:7,12 NWT
"7 On the 24th day of the 11th month, that is, the month of She'bat, in the second year of Da·ri'us, the word of Jehovah came to the prophet Zech·a·ri'ah son of Ber·e·chi'ah son of Id'do, ... 12 So the angel of Jehovah said: “O Jehovah of armies, how long will you withhold your mercy from Jerusalem and the cities of Judah, with whom you have been indignant these 70 years?"
If the “eighth month in the second year of Darius” is (October/November 520 B.C.E.), the "11th month, that is, the month of She'bat, in the second year of Da·ri'us" is 3 months later in Jan/Feb 519 BCE.
At that point Jehovah had been "indignant these 70 years" with "the cities of Judah." So Jehovah became indignant in around Jan/Feb 589 BCE, just when secular chronology places the start of siege of Jerusalem.
it-1 p. 415 Captivity
"The final capture of Jerusalem by Nebuchadnezzar was completed in 607 B.C.E., after an 18-month siege. (2Ki 25:1-4)."
So if the siege started in Jan/Feb 589 BCE and lasted 18 months, it ended in July/Aug 587.
Note that this places the start of 70 years at the beginning of the siege and ends it when God tells Zechariah : ‘“I will return to Jerusalem with mercy, and my own house will be built in her,” (Zech 1:16).