Assuredly means ‘yes, for certain’ — there is no real doubt. It is a rather formal, slightly old-fashioned way to say definitely or certainly. You will meet it more in careful writing, speeches, and stories than in everyday talk. A friend would usually say ‘definitely’ or ‘for sure’; a news reader or a novel might say ‘assuredly’. It can stand alone as an answer (‘Assuredly.’) or sit in front of a whole claim (‘This will assuredly take time.’). Though actually, if you use it too often it can sound a little stiff, as if you are trying to sound grand. It states a fact; it does not softly hope, the way surely sometimes does.