Can I bleach or otherwise whiten discolored Wii Classic Controller analog sticks?

There seems to be extensive coverage in this article:

Why Super Nintendos Lose Their Color: Plastic Discoloration in Classic Machines

Even though you are talking about the Wii, I don't believe plastic chemistry have changed significantly enough to make it irrelevant.

And the main point, with regards to bleach, is the following:

Bleach - I've had very limited success with letting bleach sit on Macintosh cases. When it does work, it's difficult to get a uniform effect unless you can soak the whole piece in bleach equally. This probably damages the structure of the plastic and might cause another type of yellowing — so beware!

A fast read through the article indicates that the change of color is caused by chemical reaction with either air or UV. Bleach will simply add another type of chemical reaction and produce uneven discoloration. Sanding a layer off seems to be one method, but that has obvious drawbacks.

The reference to retr0bright is actually Hydrogen Peroxide, of which the article states the following:

Hydrogen Peroxide (Added 02/22/2009) - In the last year or so, a new method of reversing yellowing has cropped up in some corners of the net (dubbed "Retr0bright") that involves hydrogen peroxide in high concentrations (10%-15%) and exposure to UV light.

Be warned: hydrogen peroxide in high concentrations is very dangerous (it can blind you and burn your skin), and this method can cause permanent damage to plastic if done improperly.