There are a lot of similarities in your situation to my last company, where I started at the bottom in project management and ended up CEO after a few management turnovers and the Board put me in charge. My thoughts are based on seeing cultural transition from all angles.
1. It is very, very hard to effect meaningful, long term change in the company from the bottom. What is the ownership structure of the company? Who are the managers and what is their sense of urgency in effecting change? I fear it's not much, given the situation you describe.
2. You really can't hold people accountable directly but you can try to inspire from the bottom to achieve new goals. I would not focus on "analytics" but just come up with _one_ simple, measurable goal that the team can agree on and work toward together. It's like getting a really overweight person to lose weight - start simple, with walks around the block. Once a new habit is established, move on to the next one.
3. Have you identified others in the org that share your desire to improve things, and enlisted them 1-1? Cultural/organizational change takes a lot of emotional energy. You need allies or you will go crazy.
Would be happy to get on a call and get more details and work through tactics if you are interested.