We actually have a thread discussing this very topic , so this thread might get merged into that, but it can be assumed Chai didn't tell Lan Di about the location of the Phoenix Mirror as he wanted to retrieve and give it to Lan Di himself as it would pretty much guarantee being offered a place amongst the ranks of the Chi You Men, which is Chai's greatest dream. Had Chai simply told Lan Di the location, it wouldn't have been as 'real' or important, nor would it have given Chai as much right to join the organisation, at least in his warped mind. Actions speak louder than words, I guess.

He is a fucking maniac/hardcore Lan Di fanboy, after all.

Welcome to the Dojo, dude!