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There is no such thing as a "Champion of the Enlightened". That's d20pfsrd's renaming of the Champion of Irori so as to avoid copyright infringement.
The class states that "class levels of a champion of Irori stack with monk levels for determining the effect of his AC bonus, flurry of blows, stunning fist, and unarmed strike class features".
Monastic Legacy states "Add half the levels you have in classes other than monk to your monk level to determine your effective monk level for your base unarmed strike damage. This feat does not make levels in classes other than monk count toward any other monk class features".
Just as reference for others who wish to post here.
I would say no, you don't get to double dip. Or rather, at the very least it wasn't intended to work that way.
Because of this "The class states that "class levels of a champion of Irori stack with monk levels for determining the effect of his AC bonus, flurry of blows, stunning fist, and unarmed strike class features""
So if you are level 10 monk level 10 champion of irori for the purposes of Flurry of blows, stunning fist, unarmed strike and the ac bonus" your treated as a lvl 20 monk.
If monastic legacy checks any of those qualities, it will see he has 20 levels of monk. If it raised his monk fast movement, it would see him as a lvl 10 monk.
Basically, for any of those stats the champion of irori shouldnt be considered anything other than a monk.