Of course, Pope St. Nicholas can't help but slip some incipent Ultramontanism in:
You desire to know how many patriarchs there truly are. In truth, those men should be considered patriarchs who achieve the apostolic see through the succession of bishops, i.e. those who rule over those churches which the apostles are shown to have established, namely the churches of Rome, Alexandria, and Antioch. Rome, because both princes of the holy apostles, Peter and Paul, established it by their preaching and sanctified it with their own blood which was shed for the love of Christ; Alexandria, because the evangelist Mark, who was the disciple and son by baptism of Peter, established it after being sent by Peter and dedicated it with his blood to the Lord Christ; and Antioch, because it was there that in a great assembly of the saints the faithful were first called Christians and because the blessed Peter governed it for some years before he came to Rome. The bishops of Constantinople and of Jerusalem, although they are called patriarchs, do not possess as much authority as the above [sees]. For, as regards the church of Constantinople, none of the apostles founded it nor did the synod of Nicaea, which is more venerable and celebrated than all other synods, make any mention of it; rather its bishop was given the title of patriarch more through the favor of princes than by reason, since Constantinople was called "New Rome." As for the bishop of Jerusalem, although he is both called a patriarch and should be honored as such in accordance with ancient custom and the synod of Nicaea, with his proper dignity of metropolitan preserved, in [the acts of this] same great synod, he is in no way called the bishop of Jerusalem, but rather the bishop of Helia. For on the one hand, the true Jerusalem, which is our mother, is only in heaven;[cf. Gal. 4:26] in accordance with what the Lord predicted, on the other hand, the earthly Jerusalem was destroyed by the Roman emperor Aelius Hadrian down to its foundations so that not a single stone was left atop another stone,[cf. Mt. 24:2] and it was reconstructed by this same Roman emperor Aelius Hadrian in another place so that the place of the Lord's cross, which was outside the gate, is now found within [the city] and that city is called Aelia after the aforementioned Aelius Hadrian.
 Nicholas's claim that the emperor Hadrian (AD 117-138) destroyed Jerusalem is erroneus; Titus destroyed the city in AD 70 at the end of the Jewish War.
Pope St. Nicholas also ignores that the successors of St. James the Brother of God (much like those claiming the succession to Pope St. Nicholas fled to Avignon) took refuge in Pella, and then returned. And the site of the Church of the Resurrection was incorporated into the city of Jerusalem before it was destroyed, by the walls of King Agrippa, who, as the NT tells us, dies in 44 (i.e. during the Patriarchate of St. James. Indeed, before the Council of Jerusalem).
You also ask which patriarch is second to that of Rome. According to what the holy Roman Church maintains, the canons of Nicaea indicate, the holy bishops of Rome defend, and reason itself dictates, the patriarch of Alexandria is the patriarch second to the pope.
He, of course, skips over that the canons of Constantinople I (the one whose Creed Pope St. Nicholas was seeking to replace in Bulgaria, imposing the filioque), and Chalcedon say otherwise, and the city of Constantinople didn't exist in the time of Nicea I, although the bishop of nearby Byantium attended IIRC.