A) Two-Weapon Fighting mostly has no effect on a Monk. It only applies to Weapon attacks, and Monks have none of those.
B) Weapons in your other hand do not affect your attacks, as described in the Adventurer's Vault page 56:
Weapon Properties: Many weapons have properties that provide a constant benefit. To gain the benefit of a weapon’s property, you must be wielding the weapon. Unless specified otherwise, a property affects only the weapon to which it’s attached.
So Vanguard only applies if you attack with that weapon. Actually, every weapon property should be read as "if you attack with this weapon", if it makes sense. So non-attack properties and powers are legal.
For example Weapon of Defense gives you resist all 1 when you hold it, because this property is not connected to attacking.
C) Implements do not fall under the above mentioned rule. Rain of Hammers is such a popular utility (as in not-attacking) ki focus because its property triggers no matter what you use to kill that enemy. Weapons used as implements (like heavy blades for Swordmasters) are still Weapons and are included in the rule.
As Oblivious Sage pointed out there are some ways to add damage from the off-hand weapon, the Dual Implement Spellcaster is one prominent example, but it is not really good for a Monk as it has no Arcane powers. You can get some with Hybrid or Multiclass, but it is far from optimal.
In conclulsion: Weapons in your off-hand can be really valuable, but not in attacking. They can increase your Shield bonus (Rhythm Blade), or improve your Initiative, or many other thing however.