= Solution
Form the <HNN extension> which centralizes $L$:
$$
\boxed{G_{\mathcal M}=\langle K_{\mathcal M},k\mid k^{-1}tk=t,\quad
k^{-1}r_i k=r_i,\quad k^{-1}\ell_jk=\ell_j\rangle.}
$$
This is the extension associated to the identity <isomorphism> $L\to L$; relations on the displayed finite generating set imply $k^{-1}hk=h$ for all $h\in L$. Since $K_{\mathcal M}$ is finitely presented and $I,J$ are finite, this is a <finite group presentation>.
For any $g\in K_{\mathcal M}$,
$$
k^{-1}gk=g\text{ in }G_{\mathcal M}\iff g\in L.
$$
The forward implication follows from <Britton's lemma>: if $g\notin L$, the word $k^{-1}gkg^{-1}$ is reduced with two stable letters and cannot be the identity. The reverse implication is the defining centralization relation. Consequently the computable word
$$
W(r,s)=k^{-1}t(r,s)k\,t(r,s)^{-1}
$$
satisfies
$$
\boxed{W(r,s)=1\text{ in }G_{\mathcal M}\iff(r,s)\in H_0(\mathcal M).}
$$
An algorithm for the <word problem for a group> $G_{\mathcal M}$ would therefore decide the nonrecursive set $H_0(\mathcal M)$, a contradiction. Hence $\boxed{G_{\mathcal M}\text{ has unsolvable word problem}}$.
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