Inspector Steine iwas in his office writing hisa script for thehis radio show, when a man walksed in and takesook a seat.

This just doesn't sound right in the present tense for some reason. Your sentence would work if you were writing a script and setting the scene.

After all, he is the famous journalist Harry Jupiter.

Everyone knows him unless they are idiots.

You would have to be a fool not to recognise him.

Even though Steine keeps outwardlymaintains his typical outward I-can't-be-bothered demeanour, Jupiter notices a flicker of perplexity dash across Steine’s face.

Hence, by way of explanation, Jupiter makes clearThis makes it clear to Jupiter, that he knows where he stands.

ButHowever Crystal—the murder victim—was a colleague of hims, and the Daily Clarion was demandsing a thousand-word piece byto be completed in three days.

starting a sentence with 'But' in a story is wrong most of the time. 'However' works. I keep using a different tense here, sorry I'm not sure what it's called.

Jupiter cravesis desperate to work over the details of the crime scene, which he has already gotmanaged to have a quick look at.

"And besides, you shouldn't have seen the crime scene!” Jupiter is impassivesaid, remaining expressionless.

He is still thinkings that Steine must be joking.

Far fromUnwilling to givinge up, Jupiter moves the conversation on to the “Aldersgate Stick-up” case.,

Wwhich, he claims, lies at the bottom of Crystal’s murder.

“The case was appallingly badly handled from the start.

or 'mishandled appallingly'

Then he turns to the Aldersgate Stick-up case and gets deeply irked.

and 'feels deeply disturbed'

He put it ofut of his mind a long time ago, and now it pains him to hearing about it again pains him.

Steine is unsure of how to proceeds.

A part of him wants to say with prideproudly declare that he already has an officer resolutely pursuing the Aldersgate Stick-up angle.

Inspector Steine is in his office writing his script for the radio show, when a man walks in and takes a seat.

After all, he is the famous journalist Harry Jupiter.

Everyone knows him unless they are idiotsOnly an idiot wouldn't recognize him.

Your sentence feels a little off to me, so I rearranged it a touch.

Even though Steine keeps outwardly maintains his typical I-can't-be-bothered demeanour, Jupiter notices a flicker of perplexity across Steine’s face.

Hence, by way of explanation, Jupiter makes clear that he knows where he stands.

I don't know how to fix this sentence because I don't know what you mean to say. Who are the referents for all of those "he"s? What exactly is Jupiter making clear?

But Crystal—the murder victim—was a colleague of hims, and the Daily Clarion demands a thousand-word piece bywithin three days.

So, the sooner Jupiter gets going, the better.

Jupiter craves details of the crime scene, which he has already gotten to have a quick look at.

Far from giving up, Jupiter moves the conversation on to the “Aldersgate Stick-up” case which, he claims, lies at the heart of Crystal's murder.

Which, he claims, lies at the bottom of Crystal’s murder.

“The case was handled appallingly badly handled from the start.

The culprits were never apprehended., yet it's common knowledge that Terence Chambers was the mastermind behind it.”

Yet it’s common knowledge that Terence Chambers was the mastermind behind it.” It is too much information—irritating information—for Steine to compute.

Then he turns toremembers the Aldersgate Stick-up case and gets deeply irked.

He put it offhad erased it from his mind a long time ago, and now hearing about it again pains him.

Steine is unsure how to proceeds.

A part of him wants to say with prideproudly say he already has an officer resolutely pursuing the Aldersgate Stick-up angle.

ButHowever, another part of him knows that whatever he gives away to Jupiter will end up in print.

AThis is assuming that “sure enough” is right in the first place.