Kenya had previously missed two deadlines set by the World Anti-Doping Agency (Wada) to demonstrate they were dealing with the problem effectively. A final four-week extension was given this month to approve the bill.
More than 40 Kenyan athletes have produced positive doping tests since 2012.
The new law still needs signing off by President Uhuru Kenyatta, but Sports Minister Hassan Wario is backing their chances, saying: "Olympics here we come."
The Kenyan presence on the men's podium in Olympic track events is an established tradition. Women have featured regularly in recent years.
Willis has been a vociferous anti-doping advocate throughout his career, which is set to extend to a fourth Olympics. He is satisfied Kenya's step offers more than lip service.
"The potential to impose criminal charges, fines and jail time is the strongest stance any nation has taken against drug cheats," he said.
Willis lamented the cases where athletes were drawn into the temptation of doping as a means to get out of poverty.
"Unfortunately there's not the government support for an anti-doping system, so athletes find a way through rogue doctors and pharmacists. However, I wouldn't say it was an epidemic in Kenya when compared to Russia."
Willis believed the multiple doping infractions by Russia meant they should face the same punishment as individuals for bans from the sport.
