While attempting to upgrade an environment with Windows 2008R2 + MS2010 technologies + K2 4.6.6 to K2 Build 4.6.10, I learned something about upgrading the Sharepoint 2010 piece. 

Here is the overall order for upgrading to 4.6.10.  Perform each step in order using the setup.exe file in the Installation folder in each of the steps.

Step #1.  K2 Blackpearl.  On the last page, deselect the box that says something like "Configure K2 for Sharepoint 2010" (you will do this later AFTER performing steps 2-4).

Step #2.  K2 Smartforms.  Again, deselect the check box at the end about Sharepoint 2010.

Step #3.  K2 Smartforms Control Pack

Step #4.  K2 for Sharepoint.  It will say it is installing K2 for Sharepoint 2013 but it is backward compatible.

Step #5  Once these four steps are complete, go back and do Step #1 again (select the configure option), but this time leave the checkbox selected at the end "configure K2 for SharePoint 2010".  It will take you to Sharepoint 2010 and will prompt you through the configuration Wizard for Sharepoint 2010.