Your power steering fluid is low. Have you noticed any dark spots under your car from it sitting over night?
Is there any oily residue on the bottom of your engine / engine compartment?
If so, use [url=http://www.lucasoil.com/products/display_products.sd?iid=28&catid=9&loc=show]this[/url].
If the spots under the car cease, it worked. If not and the whining does not go away, get to a mechanic.
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This is what I was thinking, but the car's only at 91k miles and I was under the impression it should have lasted longer. Is there any chance it could be low pressure in the tires? Because I haven't filled them up in ages...
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It's not the tires. Even if you were sitting on the rims the power steering unit wouldn't whine like that unless it was low on fluid. Mileage doesn't always mean anything. Chances are there is a slow leak in one of the seals or the lines. It's usually a bad seal in the rack. The Lucas stop leak reconditions worn seals and helps them re-seat. If it's a bigger leak, you likely would have noticed the car resisting turning input at the wheel some time ago.
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if it hasnt been serviced properly and frequently then small things like this will happen.