Recently Microsoft released a Windows Update Service Pack that forced Outlook 2010 to always start in Safe Mode (meaning that no add-ins are loaded; normally Safe Mode is used for debugging Outlook issues). Here you can see Outlook 2010 in Safe Mode:
Obviously, this has a big impact on the operation of your add-ins. There's a background article on this entire problem at Redmond Mag.
To fix this, the simplest thing to do is to check for updates again - Microsoft has pulled the erroneous service pack and fixed the problem in a new one.
Otherwise if you still have It is necessary to remove that service pack if you still have it. Follow these instructions:
Go to Start...Control Panel
Click on Windows Updates
Click View Update History
Click Installed Updates:
In the list of updates that appears, look for KB3114409. Note: It should be listed near the top, where the Installed On dates are Dec 9, 2015 (that is, don't bother looking past that date).
Left click the update once to highlight it
Click the Uninstall link at the top of the grid
After finishing, it may be necessary to reboot. After you restart Outlook, it will no longer to directly into Safe Mode, and your add-in will now be present.