20,000 bike commuters in Chicago times $25 = $500,000
Now, there's a lot more than 20,000 bikers in the city, but compliance won't be very high so I think the number of bike commuters, probably the most serious riders in the city, serves as a decent guide to the number of permits that will be bought.
Subtract the cost of administering the program, and the 1 hr safety class she also proposed, and I don't see this being a significant revenue generator for the city, if it doesn't lose money.
Philosophically, I'm not opposed to the idea of a city sticker for bikes, but if it won't generate revenue then there's not much point.
That's not even to mention all the extreme negative side effects of such a law, which I'm too lazy to get into right now.