Step 1
Regardless of where you are inside your Google analytics dashboard click on the Admin button on the top right section.
Step 2
Then click on the account where the profile belongs.
Step 3
Select the web property where the profile belongs.
Step 4
Click on the profile you wish to delete.
Step 5
Once inside the profile page, head over to the Profile Settings tab.
Step 6
On the lower right section, you should find the elusive Delete this profile link.