Kate, 

the lesson you have linked is referring only to transforming adjectives that already have the endings ‘- ent’ or ‘ - ant ‘ in their adjectival form. These endings are then replaced respectively by the endings ‘ - emment ‘ or ‘ - amment ‘ to transform the words to adverbs.

In the case of the adverb ‘ doucement ‘, it is formed from a regular adjective ending in ‘ e ‘ in the feminine form - ‘ douce ‘ and the rule described in link below applies : 

 Forming regular adverbs with "-ment" in French (French Adverbs)