If you have enough crafting materials and Horadric Cache materials this is by far the fastest way to gear up for a build. Since patch 2.3 you can use Kanai's cube to get legendaries faster. If I'm not mistaken the biggest Monk Community is called 'The Temple' but I'm not a 100% sure. Open the community menu and you can filter by Class. In order to find people to party with you can head to the Blizzard's forums ( European or United States depending on your server) and create a topic to find people or/and join a Class-specific (Monk in your case) community in game. Since this works both ways farming with 4 Monks should get you this set way faster. Why ? Because if you drop a piece that you already have you can share it with someone who needs it. When I was farming the Torment-only set for Demon Hunter, I've found that forming groups of 4 Demon Hunters and farm together was the most efficient. So when luck on drop is not on your side you can try. However it takes time and dedication since Kadala is not the most cooperative NPC in the game :).

You'll be assured that if you get a legendary at Kadala it's for the gear slot you are targetting and not something else. This is especially usefull when you will own several piece of the set and only a few will be missing. Slot by Slot, you'll have to spend your bloodshard at Kadala. Plus when you kill the Guardian you get some bloodshard which leads to. Nephalem rifts have a 100% buff on legendary drop chance compared to the rest of the game so if you're going to drop the set it's in a Nephalem Rift. Pick the higher Torment difficulty (important since Monkey King's Garb is a Torment-only drop) in which you can kill the mob really fast (elite pack should take less than 10 sec).

There are many ways to get legendaries but so far the fastest way I've found are.

Diablo 3 sunwuko set dungeon