I believe the full picture to be rather open, but Takayuki Kihara has shown that there is a computable embedding of $K^4$ into $K^3$ in
T. Kihara: The Brouwer invariance theorems in reverse mathematics arXiv 2002.10715
The argument is outlined in the middle of p. 13. It uses Ovrekov's "constructive map of the square into itself, which moves every constructive point", from which it follows that $\mathrm{WKL}$ fails for the computable numbers. Kihara's paper then shows that for any model of $\mathbb{R}$ in which $\mathrm{WKL}$ fails, there is a topological embedding from $\mathbb{R}^m$ into $\mathbb{R}^3$ for any $m$.
Whether we can even have a computable injection of $K^3$ into $K^2$ is listed as an open question here:
http://www.math.mi.i.nagoya-u.ac.jp/~kihara/questions.html
Although I don't have much to go on with, I suspect that asking about mutual embeddability might be the cleaner question here compared to computable homeomorphism.
