Same story here....
Went from 55 to 60
Update never finished...
Now my server service refuses to start..
Have you checked your logs \Program Files (x86)\Lansweeper\Service
Mine repeats
07/16/2014 11:47:11: 52860385 - Invalid Licence Key!
over and over again
Even though my site shows
Currently using: Premium - unlimited clients version
License valid until: 17/12/2014
As far as your issue not being able to reach the web console sounds like you have an IIS service issue...
I'd start looking there..... could have not been restarted when you applied the update.
I don't quit I just time out.......