If you really wanted to speed up this and do it as possible as possible, you can probably do this is around 3- 5 hours, if you had the perfect method to do so.
Hive majors (Earth first spawn area, 3 majors spawn, kill them then go into another and come back) Kills with sword in crucible (Take a friend into rumble and he can let you farm him for sword hilt kills which count |Or go into a game and ask a random player for help)
Class ability kills (Go to the dark beyond mission on the moon, go to the end where all the adds spawn and farm them for the kills, do not kill the knight and just farm that checkpoint) Special materials (Find one that spawns near a load area, pick it up get on your sparrow and go into another area and back and it will respawn in the same area so you can farm it) (DISCLAIMER: random players will fuck up the thing from spawning so just let them rotate to get them in a good spot again)
Wait until arms day (Just do it on the Tuesday)
Go kill alak hul (This is simple, do the strike normally then just nuke the boss, get him low then kill the warden that is the same element as the sword you want then kill the boss after) Make sure you don't fuck it up for other people. Communicate and figure who needs what.
Do all of these methods and you will get it done super easily.