We had the exact same problem last Friday. The license had expired about two months ago and we needed to apply the update with a new expiration date, but QEMC did not allow it since the service was not running.
We updated the lef.txt file located in the QlikTech folder (it can be within Program Data or Application Data, depending on the OS Version).
That solved the issue, Hope this helps.
Just a follow up on my issue.
After a tough week, we have reeinstalled everything on all of our servers - we still haven't figured out why this breakdown happened, but it seemed like something blocked for port 4747.
We suspect a windows patch and a patch from our virus software to be the reason that the services could not start.