Number of dyadic squares to cover curve of finite length

Question

I have been working on this for a while without much luck. Suppose we have a Jordan arc on the complex plane $\mathbb{C}$, i.e. the image of the unit interval $[0,1]$ under a homeomorphism, with finite length $L>0$. How many dyadic cubes of side length $2^{-k}$, for $k\in \mathbb{N}$ such that $2^{-k+1/2}<L/100$, are needed to cover the curve? I am thinking at most $2^{k}L$, but I cannot prove this rigorously (e.g. if the arc has Minkowski dimension more than $1$).