1. The first thing is to check that you have no articles with the article type specified as the referential integrity on the db won't allow a delete in this case.
2. Create a new query in query designer based on Knowledge Management - Article. Pick any attributes - I'd suggest ID and description. Ensure that on the criteria you set the article type to the one you want to delete. Set it to open the workspace article when double-clicking occurs (on last page of query config).
3. If the query returns any results change the article type to the required one (or clear the selection and click on the hyperlink to create a new one).
4. In query designer, create another new query based on Knowledge Management - Article Types. Drag on both attributes and run the query.
5. You can now right-click on the required article type and select delete. If you don't get the delete option please check your privileges or login as an administrator if you can.