Hello @Chris_Carr,
I tried to implement one way of grabbing a Case ID from the body of an inbound email and below how I imagined it 
I used a simple email including a Case ID in the body as the following:

Below, the Inbound Email Flow I used with the explanations:
The first step is to extract or select the keyword Case ID out of the body.
To do so, you can use the combination of 2 built-in functions in Architect:
FindString : returns the location of a specified occurrence of a String within a String.
Substring : gets part of a String that starts at a specified position and has the specified length.
In your example, you need to search for the keyword "Case ID: ".
At the beginnig of the flow, I created a variable called Flow.keyword that has a value of "Case ID: "
FindString(Email.Message.body,Flow.keyword)
where :
Email.Message.body is the Email body as plain text.
Returned Value is an Integer (zero-based index)

Once you have the index of the Case ID string, you can use it inside the Substring function as the following to extract only the ID :
Substring(Email.Message.body,Flow.keywordPosition+Length(Flow.keyword), 15)
This function takes 3 parameters :
String text : The source String from which the substring will be taken (Email.Message.body)
Integer start : The starting position of the String in the text parameter. The first position is 0.
Integer length : The length of the desired substring.
In our example, the Integer Start will be Flow.keywordPosition+Length(Flow.keyword)
Where :
Flow.keywordPosition : the position of the Case ID string inside the email body, then we add the length of the keyword (Case ID: ) to not include it when we grab the ID
Since you mentioned that the length of the Case ID will be always set to 15, then we put this value to make sure we cover the entire ID and not miss any digit 
That’s it !
I added a decision block to check whether the keyword Case ID is included in the body or not.
So, if the body doesn’t contain any ID, I just set the Case ID to N/A, maybe you can display it later on a script for agents.
Results:
Case ID included in the email body:
Case ID not included in the email body:
I hope that answers your question 
Kind Regards,
Charaf