I'll explain in English because I think it's more helpful considering the topic can be confusing using the same words used to describe the concept itself. 

"Meilleur" is a word that speaks to nouns. Something is good.  The noun is imbued with "goodness". 

"Mieux" is a word that speak to verbs.  Something is going well, learning well.  The action taken by the noun is good... well, the action is going well. :) 

So if a noun is described better at a specific action - mieux.  Otherwise meilleur to compare the two nouns directly.

The hint for your question is the word "bien" in the tee up. "Bien" is an adverb. So that should be your hint that we need to use the adverbial comparative "mieux" as well. 

Last point to this - in English if someone asks you how you're doing, the correct answer is "I am well" since you're speaking to a mutable condition of yourself - your health, emotional state, etc. To say "I am good" speaks to the unmutable characteristic of your "goodness".  Sounds a bit egotistical, no? 

So the use of the verb "être" does not mean to use "meilleur" or "mieux".  It's context dependent.