I'd suggest you begin by asking the most basic & difficult question:
"What problem are we trying to solve?"
Growth? Profitability? Throughput? Quality? Safety?
What are possible causes?
Malignant market forces? Core technologies at risk of becoming obsolete? Empty new product pipeline?
Decaying factories? Apathetic, stagnant or hostile work force? Dysfunctional mental models?
You can begin your analysis with analytical tools, but please, get out of your office and confirm your analysis by seeing root causes with your own eyes.
Thereby, you'll begin to develop a deeper understanding the chessboard, and of root causes and possible countermeasures.
Lean comprises an interconnected set of methodologies supported by a rich, an often paradoxical, "Thinking Way".
Lean may well be an excellent approach to dealing with your 'mess', but it's not to be entered into lightly.
If the organization isn't serious, it may be better not to embark on the journey.
Ideally, the senior leadership team and Board are fully aligned behind the transformation.
How to achieve such alignment?
Here I defer to our friend and colleague, Art Byrne, and his splendid book, The Lean Turnaround.
What if the ideal level of support is lacking? This is where things get tricky.
There should, in my view, be at least two or three respected senior champions who will support a Lean pilot in a major division.
An experiment, if you will, with the understanding that, if things go well, the organization will broaden & deepen the work.
Senior leaders must also commit to Lean education concurrent with the pilot, so they can get a basic understanding of the thinking and practices informing the pilot zone.
If the organization meets this basic test, you're probably good to go to the question of Strategy Deployment [Getting the Right Things Done]:
How will we deploy Lean in the pilot zone so as to achieve our Purpose?