= Solution
Let $f$ be the monic cubic minimal polynomial of the <algebraic integer> $\alpha$, let $\alpha_1=\alpha,\alpha_2,\alpha_3$ be its conjugates, and let $\beta_1=\beta,\beta_2$ be the two conjugates of a nonrational $\beta\in\mathcal O_K$. Because the degrees three and two are coprime, the fields are <linearly disjoint field extensions>[linearly disjoint], and
$$
m=N_{\mathbb Q(\alpha)K/\mathbb Q}(\alpha-\beta)
=\prod_{i=1}^3\prod_{j=1}^2(\alpha_i-\beta_j)
$$
is a nonzero <integer>.
Assume
$$
|\alpha-\beta|\leq H(\beta)^{-(6-\varepsilon)}.
$$
For large $H(\beta)$ this makes $\beta_1$ bounded. Since $\beta$ is a quadratic <algebraic integer>, the <height-Mahler measure formula> gives
$$
H(\beta)^2=\max(1,|\beta_1|)\max(1,|\beta_2|),
$$
and therefore $|\beta_2|\asymp H(\beta)^2$. The two remaining factors with $j=1$ are bounded, while the three factors with $j=2$ are $O(|\beta_2|)$. Consequently
$$
1\leq|m|
\leq C_{\alpha,K}|\alpha-\beta|\,|\beta_2|^3
\leq C_{\alpha,K}H(\beta)^\varepsilon.
$$
We now use the standard effective <norm form> consequence of Part (b). For
$$
L=\mathbb Q(\alpha)K,
\qquad
V=\operatorname{span}_{\mathbb Q}\{1,\alpha,\omega\},
$$
where $1,\omega$ is an integral basis of $K$, the <effective norm-form height estimate> supplies effective constants $A,C>0$, depending only on $\alpha$ and $K$, such that
$$
H(z)\leq C|N_{L/\mathbb Q}(z)|^A
$$
for every nonzero $z\in V\cap\mathcal O_L$. Its proof factors $(z)$, balances a generator using the <Dirichlet unit theorem>, and applies the <Baker lower bound for a homogeneous linear form in logarithms> to the linear relations defining $V$; the coprime degrees $3$ and $2$ exclude a unit-family degeneracy.
Apply this estimate to $z=\alpha-\beta$. Since $\beta=\alpha-z$, the height inequalities imply $H(\beta)\leq2H(\alpha)H(z)$. Hence
$$
H(\beta)
\leq C'|m|^A
\leq C''H(\beta)^{A\varepsilon}.
$$
Choose the effective value $\varepsilon=1/(2A)$. The last inequality bounds $H(\beta)$ effectively. The rational integers $\beta\in\mathbb Z$ are already covered by the stronger degree-three <Liouville approximation theorem>, and the <Northcott theorem> leaves only finitely many remaining quadratic integers of bounded height. Taking the minimum of
$$
|\alpha-\beta|H(\beta)^{6-\varepsilon}
$$
over this effective finite set gives an effective $c>0$ and proves
$$
|\alpha-\beta|\geq cH(\beta)^{-(6-\varepsilon)}
$$
for every $\beta\in\mathcal O_K$.
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