You should be checking the Distribution Service Root_date log file as that will tell you where the service is in its process of starting and shutting down as well, so if there is something there, that would be where it is being written, and depending upon where things are hanging, we would at least know how far it is getting in the process, which may then provide some additional ideas.
One likely issue would be something has become corrupted in the Distribution Service Application Data folder, and it is of little use to try to find the culprit in the majority of cases, the best course of action at that point is to stop the Distribution Service service, then rename the existing Application Data folder, then restart the service allowing it to recreate everything. The downside of this is you will lose your historical information, but you still have it in the renamed folder and you can use the Governance Dashboard application to load that up to use in analyzing things moving forward.
The only other thing of which I can think might be an issue withe Distribution Service being able to launch the QVB.exe process, which it does every time it starts, if that is the case, you could try to manually launch QVB.exe to re-register things and see if that works or not.
That is about all I have with no logs etc. shout if you have further questions.
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