The Mary’s Room thought experiment explicitly claims that Mary knows every physical fact about the given phenomenon but does at the same time implicitly suggest that some information is missing.

Not exactly. There is a sense in which seeing are red tomato conveys the same information as being told "there is a red tomato here". Nonetheless,there appears to be a difference between the two cases It is not clear whether the difference consists of some missing information,or something else.

Mary was merely able to to dissolve part of human nature by incorporating an algorithmic understanding of it. Mary wasn't able to evoke the dynamic state sequence from the human machine by computing the algorithm

There is nothing to stop Mary computing any algorithms. Humans can run through algorithms in their heads or with pencil and paper. Mary should have no problem since she is stipulated to be a super scientist. What she can't do is run the algorithm on the same hardware. She can only run it in the conscious/verbal part. not the automatic, unconscious, perceptual part.

Understanding something means to assimilate a model of what is to be understood. Understanding something completely means to be able to compute its algorithm, it means to incorporate not just a model of something, its static description, it means to become the algorithm entirely. To understand something completely means to remove its logical uncertainty by computing it.

Understanding something completely is only equivalent to understanding it's algorithm if a) it can be decomposed into a software description and a hardware platform. and b) all the significant work is being done by the software, with the hardware being just a neutral platform. Those conditions are not fulfilled in all cases.

We have seen that Mary actually can run through any algorithm, and if your expectation is that she still doesn't understand what colours look like, that would be a case where the hardware is making a difference.

You might object that Mary won't learn anything from experiencing the algorithm, but if Mary does indeed know everything about a given phenomenon then by definition she also knows how the algorithm feels from the inside.

She doesn't know everything by definition; she knows everything physical by definition. That doesn't tell us whether she would be able to figure out how things seem, phenomenally, from the inside. If we had instances of being able to successfully predict qualia from physics or neurology or whatever, there would be no need for an intuition-based parable like Mary's room. As it is. how much Mary would be able to figure out about her qualia is not something we know in advance: instead, our reaction to the story tells us what we think the answer is.