How long after a person dies does the flag remain at half-mast?

30 days. Here's Bush's presidential decree about Ford. http://www.gettysburgflag.com/FlyFlagHalfMast.php

Here's some more: Only the President and state Governors can decide when and how long the flag should be flown at half-staff. Thirty days after the death of a president or former president Ten days after the death of a vice president, the chief of justice or a retired chief of justice or the speaker of the house of representatives. Until the burial of an associate justice of the Supreme Court, secretary of a military department, a former vice president, or the governor of a state, territory, or possession. On the day of and the day after the death of a member of Congress. On Memorial Day, the day set aside to honor all the people who have died while serving the United States & originally called Decoration Day, the flag is flown at half-Staff until noon, then raised to full staff until sundown.

but you can always hang the flag upside-down to declare a state of emergency. :)

Also, the flag is only flown “half-mast” on a ship. On land it is flown “half-staff”. Sorry to nit-pick.