À qui on ne la fait plus simply means that the speakers is experienced enough to spot a trick of some sort. Isn’t there an english saying about old dogs and tricks that can be repurposed?

Le Robert réserve une entrée pour l’adjectif blasé :

Personne dont les sensations, les émotions ont perdu leur vigueur et
leur fraîcheur, qui n’éprouve plus de plaisir à rien.

Synonymes :

Indifférent, insensible, dégoûté, fatigué.

Parmi les exemples :

Blasé sur tout : être revenu de tout

et une citation que l’on pourrait retourner à votre narrateur :

“Les gens qui se disent blasés n’ont jamais rien éprouvé : la sensibilité
ne s’use pas.” (Jules Renard)

There are several difficulties in this fragment, let’s take them in order.

First, the preposition en in this context means roughly “speaking as …” (or sometimes “acting as”). It is followed by a quality which justifies the rest of the statement.

En bon pigeon, je suis victime d’une « arnaque Western Union ». = I was a regular rube and fell prey to a ‘Western Union scam’

agir en bon père de famille = to act as a good household manager, as a good providing parent

En bon syndicaliste qui se respecte, je ne peux pas laisser passer cette déclaration de notre directeur. = As a union leader, I cannot let our company chairman’s declaration pass without comment. (With the subtext that responding to the declaration is likely futile.)

En professeur de français diplômé, je lui fais remarquer qu’il a fait une faute d’orthographe. = In my quality as a certified French teacher, tell him that he made a spelling error.

This use of the preposition en calls for more than just a noun afterwards: en pigeon, en syndicaliste, etc. feel weird to me. I suspect that this is because en has many meanings and with a mere noun the phrase feels ambiguous. En bon + noun is a common combination with this meaning.

This construct usually indicates an actual quality but can also a indicate a posture.

The adjective blasé, which is used here as a noun, means someone who’s seen or practiced a lot about a particular topic. It is originally the past participle of the verb blaser. The verb etimologically meant becoming used to drinking a lot of alcohol, and generalized to becoming insensitive to anything through overexposure. The verb has become rare nowadays but the adjective is still common.

The idiom à qui on ne la fait pas means someone who will not get caught by a scam. Literally: “to which one does not make [the joke].”

Ils m’ont dit d’aller chercher la clé du champ de tir. On ne me la fait pas, celle-là. = They told me to fetch the key of the shooting range. I’m not falling for that one.

Tu ne me la fais pas. = You aren’t fooling me.

The variant on ne [me] la fait plus indicates someone who might have fallen for the trick earlier in life, but who is now sufficiently savvy to notice and avoid it.

Thus à qui on ne la fait plus does barely more than reinforce blasé, which is necessary for the use of en blasé … to be understood.

En blasé à qui on ne la fait plus, je réponds …
I haven’t falled for that one in over thirty years, so I answer … [I caught the trick]
Taking my most jaded air, I answer … [I’m trying to pass off as jaded, but I may be in over my head]