Does it work?
When making a full-attack action, the wielder of a speed weapon may make one extra attack with it. The attack uses the wielder's full base attack bonus, plus any modifiers appropriate to the situation. (This benefit is not cumulative with similar effects, such as a haste spell.)
Once per round, the wearer may add an enhancement bonus of +1 to +5 on one attack and damage roll for an unarmed strike or natural attack (for one specific attack, not all attacks made with an unarmed strike that round). The wearer may use this item an additional time per round when his BAB reaches +6, +11, and +16.
Choosing to enhance an unarmed strike is not an action and may occur when it is not the wearer’s turn (such as when making an attack of opportunity). The wearer must decide to use the item before the attack roll is made, but does not have to expend all uses at the same time. For example, if the wearer can use the item twice per round, he can use it once on his turn when making an attack and save the second for the possibility of making an attack of opportunity.
Additionally, the bodywrap can grant melee weapon special abilities to a creature’s unarmed attacks, so long as those special abilities to be added apply to unarmed attacks. See Table: Melee Weapon Special Abilities for a list of abilities. Special abilities count as additional bonuses for determining the market value of the item, but do not modify attack or damage bonuses. Any special abilities are set at the time of creation. A bodywrap of mighty strikes cannot have a modified bonus (enhancement bonus plus special ability bonus equivalents) higher than +7. Unlike an amulet of mighty fists, a bodywrap needs to have a +1 enhancement bonus to grant a melee weapon special ability.
Would I get the extra attack if I only enhance one of my unarmed strikes with the speed property? Alternatively, would I need to enhance one normal unarmed strike and also apply the property to the extra unarmed strike from the speed property to make it work?
You only need to enhance one in order to have the speed property function. For instance, two-weapon fighters only need to have a single speed weapon to gain a hasted attack even though they are fighting with both weapons. So yes, this works.
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The speed property triggers when you use a weapon with that property in a full attack. So here's how it works:
1. Declare a full attack.
2. Make an attack with a weapon with the speed property, or in the unique case of the body wraps, you apply the speed property to the an attack as you make it.
3. Check whether you have already gotten an extra attack from any other speed weapon or haste effect. If you have, do not proceed to step 4.
4. The triggers for the speed weapon property are fulfilled, giving you an extra attack with that weapon. It is not relevant that it does not have the speed property for the additional attack, as long as it is the same weapon.
Let it never be said that paizo does not give monks nice things.