After interviewing the bank manager, Constable Twitten asked for a place where he could ponder over the investigation without being interrupted.

Being always so thorough, Twitten assessed every detail, evidence and clue, trying to make sense of them.

After three, intense hours, Twitten had finally worked out who the culprit was.

You'd only use had for an action before another action in the past. For example, "Twitten had worked out who the culprit was before he finished his coffee."

CravingEager to make his astounding report to his senior officers, Twitten stood up and went over to the door.

After interviewing the bank manager, Constable Twitten asked for a place where he could ponder over the investigation without being interrupted.

Usually one would only say ponder, not ponder over.

The manager offerred his own office, and Twitten accepted.

two very minor errors here:
I'm not sure if this is just a type, but the correct spelling is "offered," not "offerred." I don't know if there is exactly a reason why-- it's just one of those things that you learn over time.

Technically, when using a coordinating conjunction (for, and, but, or, etc.) to connect two phrases that could stand alone as sentences (in this case "The manager offered his own office" and "Twitten accepted"), a comma is supposed to be used before the conjunction. I wouldn't worry too much about this, though, as even many natives will make this error regularly.

Twitten was resolved to get to the bottom of Crystal’s murder (case) for good, so nothing must get in his way now.

It seems a little unusual to include the word case. The reader would already understand what you meant from the word "murder" on its own, so no additional specification is needed.

Being aAlways so thorough, Twitten assessed every detail, evidence and clue, trying to make sense of them.

This sentence is totally passable as it is, but it would sound more natural to drop the word "being" and just begin the sentence with "always so thorough."

After three, intense hours, Twitten had finally worked out who the culprit was.

Although you often need a comma between two adjectives, in this case it is not necessary. I believe this is because the first word is a number, so it isn't a conventional adjective.

Mrs. Groynes, the police station cleaner, was standing in the doorway with a gun in her hand.

For short titles like "Mrs." "Mr." and "Dr." a period is always put at the end of the word to signify that it is abbreviated.

After all, he had just worked out who Mrs. Groynes really was, so he wasn’t bewildered ato seeing her pointing a gun at him.

"Bewildered at seeing" is understandable but a little unnatural. The word "at" isn't usually used after "bewildered" and instead one would say "bewildered to see."