(This is not a complete answer, but I cannot comment.)

It follows immediately from the main result of

http://arxiv.org/pdf/1203.0686.pdf

that for every $\alpha<k/n$ there is an $\alpha$-Hölder map from $I^k$ onto $I^n$.

In the general metric space case of that theorem, a map with optimal Hölder exponent need not exist.

In this very specific case, I would bet that it does, but I don't have a specific construction in mind.