Login to your MadMimi Account. If you do not have any account already, signup for a FREE account here
Once logged in, hover over Account on top right
Click Account on the top menu
In Account Details page, look for API tab on far right
This shows the API key you need.
If you do not see your API key, click Regenerate API Key button.
Copy the API key
In your WordPress website, go to WP-Admin > Opt-in Samurai > 3rd Party Integration
Click Activate button under MadMimi Logo
Paste your API Key in the form in bottom
Hit Save Changes button and you are done!