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We need to think about how to best set up the HMM. One of the important questions concern continuity: since we have around 10 minutes of observations per day, the observations are not continuously connected.
Some possible solutions:

• Assume states do not change more than once during the unobserved intervals and pretend the states are in fact connected. Pro: simple; con: probably not entirely realistic, will lead to skewed estimates of state durations
• Instead of using the transition matrix $T$, use $T^K$, where $K$ is the number of unobserved time steps. Pro: more realistic; con: probably quite hard to do inference in practice, because the state probabilities become very small for most transitions
• Consider all observations as independent (unconnected) time series that share their transition matrix; Pro: probably more realistic; Con: observations might only contain between 1 and 5 data points, which means few transitions will ever be observed, and most transitions will be between the same states
• Full Bayesian inference: Can we integrate out the missing time steps entirely?

Here is a useful HMM reference.