Currently my organization is in a very similar situation. Apparently before my arrival an attempt was made to upgrade in place from 2008 to 2012r2, and it seems to have failed. So you may not be able to do it that way. The advise I was given was migration of Database from SQL 2008 to SQL 2014 (not 2016 though I believe 2012 would probably work), I was advised an in place upgrade from 2012 to 2016 (for the App/Web components) would be possible but that's something I have not yet attempted. Currently migrating our Dev environment from 4.6.11 (Win 2008 Servers) to K2 4.7 (Win 2k16 with SQL 14 and Win 2k12 for App/Web as 4.6.11 does not support 2016). Just my grain of salt, hopefully you find it useful.
We are working in the same thing...
3 Environments DEV/TEST/PROD all running K2 Black Pearl 4.6.11
Currently working in a new state upgrade machine. just to make sure everything is ok.
Game plan -> Package all apps from o 4.6.11 - DEV / TEST / PROD ] -> Deploy in Stage and Perform In Place upgrade. and see what breaks...
later plan to do Inplace upgrade to K2 Five..
I really wish a much better way of doing this =(