What do you advice a leader to build trust, empowerment, within a team when failure happens?

Extend unconditional support-- take responsibilities for the team performance-- Bring forth innovative ideas and thinking that can easily accommodate with the existing line of operations to bring convincing results-----here actions are more powerful than any words or console.

Recalculate the project, figure out what went wrong and talk about it. Then reassign the project and collaborate until progress has been made. 

You have to go back to the good basic fundamentals, not to reinvent the wheel but to go back to the good manners, visions, missions, goals, objectives, planning,..............

gently advise him /her learn from failure, we are all team, work together for acheivement of goals and success.

A team will revive from its failure quickly, when the team not been accusing anyone or each other about failure, destructive or personified feedback.

Leader should setup a culture to provide them collective learning out of what is important and set the team to move forward..

It has to be only feed forward and not of feedback, especially when team faces failure. 

Recommend you to watch Marshall goldsmith video on Feedforward...