The best way to vary the beginnings of sentences 3-5 is Kingsley's family expected her to stay home to care for her mother and younger brother. Her mother was sick at the time. In 1892, her parents died.
These are known to be  words or phrases that helps a person to introduce the other part of a of the sentence.
Note that  The best way to vary the beginnings of sentences 3-5 is Kingsley's family expected her to stay home to care for her mother and younger brother. Her mother was sick at the time. In 1892, her parents died because it set apart the events by commas and tells what reader what to expect.
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