There are some references for computing the Hilbert Class Fields of quadratic extensions, and of cubic extensions.
Q1. Is there a general way to compute the Hilbert Class Fields of cyclotomic extensions? (i.e. HCF of $k(\zeta_n)$, when HCF($k$) is known.)
Also it is known that if $K$ is the Hilbert Class Field of $k$ and $L/k$ is finite Abelian, then $KL$ is contained in the Hilbert Class Field of $L$.
Q2. How far is the converse true? I.e., if $L/k$ is finite Abelian, and $M$ is the HCF($L$), then can anything (non-trivial) be said about HCF($k$)?
Edit: Barry Smith's comment suggested that, in order to solve this, we would first need to know:
Q1.5. How far is the HCF of $L$ from just $KL$?