description of continuous probability distribution
You're confusing the Shannon entropy of a discrete probability distribution with the differential entropy of a continuous probability distribution. The minimum distribution length is only given by the Shannon entropy for discrete probability distributions.

What the fact that the differential entropy is 0 for this probability distribution means is that for large $k$, you can approximate this random variable with an expected error of $1/2^k$ using $k$ bits. If the differential entropy were $\ell$, you could approximate it with an error of $1/2^k$ using $k+\ell$ bits.