The controlling language of Article V, Section 7 of the Illinois Constitution states:
...The appointee shall hold office until the elected officer qualifies or until a successor is elected and qualified as may be provided by law...
The vacancy occurred upon Topinka's death, as did Quinn's right to appoint a replacement. According to this language, the appointee serves until an elected successor qualifies. So if Sheila Simon had won, the appointee would have served until Simon took the oath. With Topinka unable to take the oath, Quinn's appointee is not divested of office until a special election is held and the winner takes the oath.
That's my take and the issue doesn't seem that hard. But this is Illinois.
Of course they do, the term is going to EXPIRE.