your proof seems fine to me. one caveat, there is nothing saying L might not EQUAL K, so you need a one-line dispatch for that. the degree of L over K isn't relevant, and without it given, you can't say much about the degree of the minimal polynomial of r.
as for a counter-example, consider , which is isomorphic to . this is certainly a subring of , which contains . however, it is NOT a field, because (for one), it doesn't contain