Here are a few links to help.

Office 365 Desktop Setup Tool for Office 2010 and Office 2007

https://www.microsoft.com/en-us/download/details.aspx?id=50360

And or.

Service Pack 2 for Microsoft Office 2010 (KB2687455) 32-Bit Edition

https://www.microsoft.com/en-us/download/details.aspx?id=39667

This will fix 90% of issue with outlook 2010 prompting for a password over and over again even though you have it right