How to prove the language of suffixes of a regular language is still regular?
Your conclusion that $L’$ is a concatenation of two regular languages is wrong.

$L’$ is basically the set of suffixes of all the strings in $L$. Try to prove it for the prefixes and use the “closed under reverse operation” property of regular languages.

Hint for prefix language: make some special states as final state in the DFA of a regular language.