I put "that old patient became ill last week" and got it wrong - is there a nuance I've missed here?
"a week ago" and "last week" are not quite the same - one uses a duration and one identifies a particular timespan.
Also, consider someone who became ill on Friday and you're talking on Monday. He got sick last week, not not a week ago.
got it - so how would I say "last week"?
I would say, "la pasintan semajnon". (With the accusative of time.)
Ankaŭ vi povas diri ke oni pasintsemajne malsaniĝis
"That elderly patient got sick a week ago." not accepted
Last week covers anytime in the seven days consisting of last week.
A week ago is only one day, seven days ago.