Is there some sort of pattern detecting thing, whose name perhaps includes something like "markov" or "kolmogorov" or "bayesian", that could automatically take a time series data and predict the next values based on an unknown, complex model?

Prediction isn't magic -- you need relevant information. In this case, if information about the causes of your sleep patterns are not in the time series, you won't be able to accomplish much with this approach.

I note that you include info about caffeine and other drug usage in your data. These are obvious possible causal agents, so it's necessary but not sufficient to have information about them. I suggest the following approach:

1. Investigate (i.e., do a literature search on) possible time-varying causes of sleep disturbance.
2. Collect months-long time series data on these potential causal factors, concurrent with time series data on your sleep patterns.
3. During your waking hours, study causal time series analysis.
4. Analyze your data and intervene on any identified causes.
5. (Optional) Write it up for LW or for even as a journal article.

Comments:

On point 1, I'd say the odds are better than 50:50 that you've already done the relevant lit review. I still included it in the plan of attack in case you hadn't.

On point 2, I suggest months-long exhaustive data collection only because you've already shown that you are motivated to do it. Also, I think it's important to collect data on your eating habits whether or not it shows up as potentially causal in the lit review.