How come enlightened people don't have any laziness? I just heard that enlightened people have zero laziness. How is it possible? What is "laziness" ? What is the connection between laziness and ego/desire? 

Edit: Also that monk said: "Laziness is caused by ignorance."
Speaking from the first hand experience, laziness comes from attachment, fear, or ignorance. No attachment, no fear, and no ignorance - translates to no laziness. 

And how does attachment cause laziness? Someone is attached to what they think is a pleasant pastime, taking delight and enjoying that experience. For such person, anything apart from that experience is unpleasant, hence they will have inner resistance and avoidance: laziness.

And how does fear cause laziness? Someone is afraid of failure or is afraid of judgement or is afraid of difficulties, or is afraid of all three. For such person, doing anything that has a risk of the three looks dangerous, hence they will have inner resistance and avoidance: laziness.

And how does ignorance cause laziness? Someone is confused about what is important, confused about what's good and what's bad for oneself and others, confused what actions lead to what outcome. For such person there's no understanding: "this needs to be done". Having no understanding, they have no motivation. Having no motivation, they are lazy.

An enlightened one, having no attachment, taking no delight in pleasant experiences, having overcome the sign of "pleasant", having no fear, having no resistance to difficult experiences, having transcended the notion of "difficult" - does not have a basis for the arising of inner resistance and avoidance, therefore laziness can't arise. Having the perfect understanding of what's good and what's bad, knowing what is important, an enlightened one acts with wisdom and compassion.