Complex sentence: Via Lady Sidney Elizabeth discreetly indicated her earnest interest in marrying the Archduke--when she cooled down again within days, Mary Sidney felt betrayed and was angry at her brother and the Queen.
Split into simpler sentences: Via Lady Sidney Elizabeth discreetly indicated her earnest interest in marrying the Archduke. She cooled down within days, however, as a result of which Mary Sidney felt betrayed and was angry at her brother and the Queen.