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This might be incorrect.

Using the formula: Casting time = spell cast * (1-[Fast Cast]/100),
I get these values for Utsesumi: Ichi (a 4 second spell cast):

32 Fast Cast = 2.72 (Everything listed)
27 Fast Cast = 2.92 (Everything but the MKD helm)
22 Fast Cast = 3.12 (Everything but the katana)
17 Fast Cast = 3.32 (Neck, body, earring only)

I haven't taken other spells into account since you'll usually only cast debuffs/elemental ninjutsu when you have Utsesumi up, and since Utsesumi Ni already casts fast enough to go off between monster swings. For the sake of the calculations, I've assumed the neck piece just works like normal fast cast.

A pretty standard delay for monsters is 180 (I think), meaning they'll attack once every three seconds. Hojo: Ni, when unresisted, increases this by a factor of 200/1024 (there seem to be some inconsistencies in the reported value of hojo, but the 200/1024 value is the only one I found that had testing attached) to ~215.15, meaning they'll attack once every ~3.58 seconds.

If this was done correctly, it suggests that there's no need to equip anything more than the Body, Earring, and Neck in order to cast Utsesumi: Ichi between swings even if you're soloing. In fact, you could do away with either the body or the earring and still make it, but it would be a lot closer and those pieces are pretty easy to obtain.

My rationale for trying to exclude the helm and katana is as follows: The helm needs to be switched to a haste helm midcast in order to cap haste (without the ring), which is fine if it works but if you're lagging a little you're going to miss out on a lot of haste, and the katana is already outclassed by other weapons.

The only good reason I can think for using the katana is if you need to keep Migawari up as much as possible (soloing Indrik).

Another possible use is if you find yourself fighting a monster which is naturally resistant to slow, or if you can't land hojo. With 27 fast cast you should still be able to sneak an Ichi in between casts. You'll get hit if you're a little slow hitting your macro, but it should be easy to time it so that the hit lands early enough in the cast that you go uninterrupted. Casting Ni -> Ichi properly will avoid this altogether, though.

A third use is if you find yourself tanking something that hits 100% of the time, where you need to reduce your timers as much as possible (and where you'll presumably switch to either a -DMG taken or evasion following your cast). This was more common before abyssea (sub DRK tanking, etc.). It should be noted that the helm does not apply to the last situation -- fast cast pieces have to remain equipped throughout the casting of the spell in order to reduce the recast timer. You're better off keeping your haste helm equipped.

Summary!
The Neck, Body, and Earring are enough to cover you for most situations!
The katana's use is very situational.
Save your MKD helm augment for something else!

Hopefully I didn't mess the calculations up.

Edit: After rereading this, I think I need to clarify a point regarding the use of the katana. Since the fast cast is not needed in most situations, its only use is the recast. There are too many situations for a blanket response to cover, but I will say this: I've never had trouble with recasts inside of abyssea, except for the few times I've tanked an Ironclad. Outside of abyssea you're probably better off using an evasion katana if your shadows are dropping too quickly.

According to ffxiclopedia, Anwig Salade can be augmented with 5 Fast Cast. The blood pact option gives +3.