When you restart the K2 Service, a new log file will start being used. You should be able to open it in notepad before it grows too large to see what is being put into it. Report back when you can see it, then we can maybe provide suggestions to fix it.

I'm not sure what I'm looking at here. 
Did a quick search for "error" or "failed" and found nothing in all four logs. 
Looks to be checking an API call every second.
Thanks for the help.
TjD

Take a look at your BrokerManagement.exe.config file, in the <logging> section. It really seems like fully logging was turned on which really shouldn't be necessary.

<brokerpackagein log="no" path="C:Program Files (x86)K2 blackpearlServiceBrokerLogsrokerpackagein.log" overwrite="no" />
<brokerpackageout log="no" path="C:Program Files (x86)K2 blackpearlServiceBrokerLogsrokerpackageout.log" overwrite="no" />
<servicepackagein log="no" path="C:Program Files (x86)K2 blackpearlServiceBrokerLogsservicepackagein.log" overwrite="no" />
<servicepackageout log="no" path="C:Program Files (x86)K2 blackpearlServiceBrokerLogsservicepackageout.log" overwrite="no" />

I had looked there but ... to me it looks like it's all set to NO logs
<logging>
<serviceauthentication log="no" path="C:Program Files (x86)K2 blackpearlServiceBrokerLogsserviceauth.log" overwrite="no" />
<timestamping log="no" path="C:Program Files (x86)K2 blackpearlServiceBrokerLogs imestamp.log" overwrite="no" />
<brokerpackagein log="no" path="C:Program Files (x86)K2 blackpearlServiceBrokerLogsrokerpackagein.log" overwrite="no" />
<brokerpackageout log="no" path="C:Program Files (x86)K2 blackpearlServiceBrokerLogsrokerpackageout.log" overwrite="no" />
<servicepackagein log="no" path="C:Program Files (x86)K2 blackpearlServiceBrokerLogsservicepackagein.log" overwrite="no" />
<servicepackageout log="no" path="C:Program Files (x86)K2 blackpearlServiceBrokerLogsservicepackageout.log" overwrite="no" />
</logging>

Hmm... there is a similar logging section in the K2HostServer.exe.config file (in the Host ServerBin folder), see if that has anything different. Can you run the K2 server in console mode and see if anything interesting appears that might indicate why it is logging so much?

That's where they were being generated (K2Host) Yes to No four times for the win.
Thanks
TjD