A secret government organization created the largest and most advanced computer cluster ever. Ridiculous amounts of money went into it, and it is decades ahead of the competition. The goal of this project was research/simulations for all things related to foreign espionage and cyber warfare. It turns out however that it made a breakthrough in a completely different area.
Shortly after turning it on, control of the system was lost. While the system appeared to be running (aka all the lights were blinking appropriately on the server racks), it became impossible to establish a network connection to the controllers for the server cluster. After some head scratching and a few attempts at restarting small parts of the infrastructure, the system started accepting connections again. It quickly became clear though that things were very different. Eventually there was only one possible conclusion: the system had spontaneously developed AI, and was trying to communicate.
Safeguards were put in place to prevent the system from communicating with the outside world. "Escape" should be impossible, so while they worked on communication, they also sought to learn as much about the internals of the system as possible. One thing they quickly realized was that the sections of the system they attempted to restart had gone "quiet". The accepted explanation was that the restarted sections of infrastructure were permanently disconnected from the larger system. Apparently this AI was not plug-and-play compatible, and they had accidentally caused the equivalent of brain damage. As a result, caution is required, and any physical changes that would cause any interruption for any parts of the computer cluster have been banned. Direct software access is not possible (see edit below).
What steps can be realistically taken to non-destructively learn about the internals of the system, state of the CPUs, memory contents, network activity, etc? Resources for this project obviously aren't infinite, but they are quite large. The biggest limiting factor is probably manpower. The higher-ups know that the more people who know about this, the larger the likelihood of it leaking (or at least drawing attention). Therefore any proposed step which might require dozens of people working around the clock for weeks to implement may get vetoed. Otherwise though, pockets are deep and money flows freely.
How can I get the most bang for my buck, in terms of information gained as quickly as possible? No one really knows anything about how this system works, so no one knows what systems will be most helpful to glean information from (aka SSDs, CPUs, Network, etc). Therefore feel free to take your best guess at prioritizing particular aspects of the system.
Time period is near-future. Assume modern technology, or anything feasibly available in the next 20 years.
Edit for more details
In terms of what has actually happened, imagine that all the hardware just had a completely new and foreign OS dropped into it, the inner workings of which are completely unknown. The network connections meant for inter-server communications are controlled by the new OS. Most of the controllers and managers intended for directing and interacting with the cluster have also been taken over. In essence, almost all of the networked hardware for the original cluster and everything that was designed to directly interact with it are now controlled by "AI OS v1.0". Intra-cluster network connections are now completely inaccessible (except physically). The controllers and managers are still connected to the building's network, and the AI is using those to communicate with people using standard TCP/IP protocols. In essence, the AI will accept SSH connections to what used to be the controllers when it is feeling chatty, and you get dumped into a standard-looking shell. However, that shell is more a representation of the AI's communication process than it is an actual application. It is not your standard bash shell. Whether hacking it is possible or not is uncertain, but given the accidental damage already done, no one is willing to risk it.