This implies that now that his sister has recovered, her brother decided/chose to come down with ... (you don't state whether it is the same illness or some other illness)
the meaning of 'as' in this sentence = "because" because of the tense you have used: "But as she had recovered..."
that is, because this happened (she recovered), therefore as a direct result, this (her brother falling ill) happened.
Perhaps he wanted some of the attention lavished on his sister whilst she was feeling so ill. ! :lol: