How do I run my automator service without opening automator?
In summary, to create a keyboard shortcut to run your Automator service you will need to use:


Automator to create your service so that it'll work the way you want
System Preferences to assign a keyboard shortcut to the service


Below are the steps involved for each. 

1. Create a service using Automator

You're on the right track with your Service, however you need to change where the Service receives no input from. Below are the instructions (from scratch so that it's useful to others):


Launch Automator (usually found within your Applications folder)
Go to File > New
Select Service and click Choose
In the top right hand of the window that appears, ensure that "No input" is selected from the Service receives drop-down list
Ensure the In drop-down list is showing "Any application" (Note: This is where you went wrong as you had selected Finder)
In the second column containing the long list, find "Get Specified Finder Items" and double-click on it
This will add the Get Specified Finder Items window on the right
Now click on the Add... button to select the design kit folder you're wanting to copy
Once again in the second column containing the long list, find "Copy Finder Items" and double-click on it
Save the service using a meaningful name (e.g. Duplicate Design Kit).


Now to the next step.

2. Creating your shortcut


Go to System Preferences > Keyboard > Shortcuts
Select Services from the sidebar
Find your service (it'll be in the list on the right-hand side)
Add a shortcut by double clicking on the service name
Now go to System Preferences > Security &amp; Privacy > Privacy
Select Accessibility in the sidebar
Click on the + sign (you may need to unlock the padlock if it's locked)
Add Automator
Add Finder (to find this you will need to navigate to /System/Library/CoreServices/Finder.app).
Exit your System Preferences.


Now you should be able to do two things:

1. Run the service from any application by going to the Services list within any Application menu (e.g. Finder > Services, Safari > Services, TextEdit > Services, etc) and select the service you just created. 

2. Use the keyboard shortcut to run the service.

Let me know how you go.

[UPDATE]

For an unknown reason the OP found that he needed to restart his Mac before the Service was available. I mention this here in case it helps someone else trying to achieve the same thing.