She's the head of state, not the head of government. She doesn't have the power even theoretically to make any executive or legislative decisions. Aside from "representing" Canada in a very vague sense, the closest thing to real power that she or her representative (the governor general) has is to "decide" which party to ask to attempt to form a government after an election. This is basically always simply the part with the most seats, barring very rare exceptions.

Anyhow, tl; dr: Canada is most certainly independent, and gained its independence peacefully (although there were minor conflicts prior to that).