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Rand believed that you could determine the proper rules for society by a simple rational process.

Actually, it was more of a logical process, in that she had embedded assumptions from which she drew conclusions. However, there is no rational basis on which one can prove those initial assumptions. (That's what makes them assumptions.)

If you accept as stated in the article that Rand's objectivist philosophy is: 'Good = that which is free, productive, etc.' then you may draw logical conclusions from that premise. But what if 'Good = that which provides the longest, healthiest life for the most people'? Or, if one accepts the existence of a higher power than mankind (which Rand rejects), then the proper premise is: 'Good = that which helps man to achieve Salvation/Paradise.'

In fact, Rand's philosophy as expressed in 'Atlas Shrugged' is inherently illogical. Those who 'shrug' off the burden of the unproductive (giving up their productive wealth and power to withdraw to the mountain hideaway) in fact are making a sacrifice which will (ultimately) benefit others - which is doing exactly what the unproductive want. It's a different group who benefit (perhaps the descendents of those who 'shrug' off their burden instead of the multi-generation unproductive) but it's still sacrificing for others. The ultimate example of this is when John Galt tells the torturer how to fix the machine they are using to torture him. He sacrifices so that they may achieve their desire, even though that desire is undeserving. Shouldering the burdens of the unproductive/incompetent.

And in that way 'Atlas Shrugged' and 'Lord of the Rings' do show a strong parallel. In each case, the fight against evil requires self-sacrifice.

Rand's primary problem is that she had no faith in anyone or anything higher than herself. Her ego got in the way of accepting faith as a 'rational' basis for action - yet it is an inherent aspect of the human animal to have faith in things not provable (for example: Love) and she based her own philosophy on faith in a particular premise on the definition of good and evil. She rejected part of humanity in an attempt to show what she considered the highest form of humanity, and so - as you put it so plainly - she was wrong.