I would assume that the answer is yes, because $K \subset L$. Isomorphism implies that the cardinalities are the same, and so $L \subset K$.
But if the answer were so simple, then it would not exist as a question in a text, so I am wondering if my argument is incorrect.
I'm assuming that "$L/K\cong K$" means "$L$ is a field extension of $K$ which is isomorphic to $K$."
Remember that infinite sets can have proper subsets of the same cardinality (indeed, that's sort of their defining feature). For example, take the natural numbers vs. the natural numbers greater than $1$. So your argument breaks down.
This is something you'll want to exploit: whip up a field extension $L\subsetneq K$ where the relationship between $L$ and $K$ is similar to the relationship between two infinite sets of the same cardinality. HINT: Think about fields generated by adding, to a given starting field, a bunch of "indeterminates" ...