The article very clearly notes that the lending program is only available on actual Kindles. It even specifically calls out the fact that applications on other platforms (like the iPad) do not include this feature.

Amazon must have concluded that people with Kindle hardware would be more likely to buy books than those with who have Prime but an iPad which uses the Kindle, Nook, and iBook stores to name a few.

Maybe they'll loosen that restriction after the holidays when the free Prime trials from the initial round of Kindle Fire sales will have expired, potentially adding many new Prime subscribers.