Steven, Congrats on completing this research and your conclusions. I only have one small problem with it. The definition of Takt time used in the Six Sigma world is not something you calculate from your process but it is Defined by your customer demand for your product or service. Takt time is the ‘Demand time’….the time that you IDEALLY will take to turn out each consecutive product or service in order to keep up with customer demand for that products or service. The ideal CYCLE Time is one that is Equal to TAKT Time. This will make then stop both delays for the customer and also stop over-production. Correct me if I am wrong, but I don’t think this definition or Cyle and Takt time appears in your analysis ? Thanks. John Dennis
Cycle time variation is a metric and philosophy for continuous improvement with the aim of driving down the deviations in the time it takes to produce successive units on a production line.  It supports organizations' application of lean manufacturing or lean production by eliminating wasteful expenditure of resources. It is distinguished from some of the more common applications by its different focus of creating a structure for progressively reducing the sources of internal variation that leads to workarounds and disruption causing these wastes to accumulate in the first place. Although it is often used as an indicator of lean progress, its use promotes a structured approach to reducing disruption that impacts efficiency, quality, and value. 
Without seeing how complex the information is that you have collected it can be hard to give a definitive answer to your question. But from what you say a VSM could be one way to show where information is being held up if you show the workload that is at each stage in the process; maybe with a work in progress measure showing the number of documents with an average time for processing each.
Be careful when measuring processing time, maybe it only takes a few seconds to sign a piece of paper but they will often sit there for days before being reviewed. I would be inclined to take an average number of documents processed in a specific day to calculate a processing time.
I once did a similar process improvement on the flow of paperwork through a company; this was done as a “brown paper” exercise which took up a whole wall of a large meeting room to map out the flow. Despite showing major hold ups people were still resistant to change – that was until we turned the company CEO into a “document” and walked him step by step through the process, desk to desk, explaining how long he would sit on each desk etc. After the initial explosion and fall out things were soon improved. People were given responsibilities to approve work rather than sending multiple documents through many different departments and other issues.
Best of luck with what you are doing.