Question about $\sum_{k=0}^\infty \frac{1}{(H_{2^k})^x}$

Question

Define a function:

$$f(x)=\sum_{k=0}^\infty \frac{1}{\bigg(\sum_{n=1}^{{2^k}}\frac{1}{n}\bigg)^x}=\sum_{k=0}^\infty \frac{1}{(H_{2^k})^x} $$

where $H_k$ is the k-th harmonic number.

Questions:

1. For what values of $x$ is $f$ convergent?

2. Can $f$ be analytically continued?

I think $f$ converges for $\Re(x)>1.$ And I think it can be analytically continued but may not have a nice closed form in terms of known functions.