There are many ways to get legendaries but so far the fastest way I've found are...
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). Then go rifting. 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. Plus when you kill the Guardian you get some bloodshard which leads to...
Slot by Slot, you'll have to spend your bloodshard at Kadala. This is especially usefull when you will own several piece of the set and only a few will be missing. 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. However it takes time and dedication since Kadala is not the most cooperative NPC in the game :).
So when luck on drop is not on your side you can try...
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. Why ? Because if you drop a piece that you already have you can share it with someone who needs it. Since this works both ways farming with 4 Monks should get you this set way faster. 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. Open the community menu and you can filter by Class. If I'm not mistaken the biggest Monk Community is called "The Temple" but I'm not a 100% sure. However you should find quite easily.
Using Kanai's Cube
Since patch 2.3 you can use Kanai's cube to get legendaries faster. If you have enough crafting materials and Horadric Cache materials this is by far the fastest way to gear up for a build. Basically you simply have to :
1) Use the Receipe : Hope of Cain to upgrade a level 70 rares items into legendaries until you get the legendary you need. If you need rares items you can either :
- Gamble at Kadala and upgrade the rares she gives you.
- Craft level 70 rare of the desired type.
2) Once you have your desired legendary, use the receipe Law of Kulle until you get the desired affixes.
If you need to finish a set, you can also use the receipe Skill of Nilfur that allows you to swap an item from a set to another random item from that set. Then use the Law of Kulle to get the desired affixes.
Using this method I've been able to fully gear my last Monk in less than a day. Of course I had prior farmed a lot of crafting and Horadric Caches materials.
Note that both upgrading a rare and rerolling with the law of Kulle can result in getting an ancient legendary. However this works both way and rerolling an ancient can transform it into a regular legendary