When the server died, it may have left 2 tables in one of the sql databases behind in a corrupt state, which lead to (at least) M-Browser and the Stats not working.
I hadn’t noticed that when I brought the server back up, I tested M-Browser etc., so it did work after I had brought the server back up.
2-3 hours after the server went back online – as per the computers event log – those 2 tables couldn’t be read by the sql server.
I have now repaired them and everything is working again.

BTW, M-Browser can be used offline when you run it like this:
…\Kingpin\Kingpin_M_Browser.exe o
If it cannot connect to my server, it will fall back to that mode automatically.
It didn’t in this case, because the client was actually able to connect, only that the request couldn’t be processed.

When M-Browser is run in offline mode, it displays a number of kp servers which are hardcoded in the client, without checking whether or not the servers are online.