Q: How to use Bootcamp with a broken optical drive?

Did the OD stop working because of Yosemite? BCA can tell you what versions of Windows are allowed when you start BCA.

Here is a list of Mac Model Identifiers which are Win7 only.

      <key>Win7OnlyModels</key>

        <array>

                <string>MacBook7,1</string>

                <string>MacBookAir3,2</string>

                <string>MacBookPro5,5</string>

                <string>MacPro2,1</string>

                <string>Macmini4,1</string>

                <string>iMac10,1</string>

        </array>

How to identify iMac models - Apple Support

2010 iMacs are iMac 11,x models.

The reason it stops working is because a certificate tied to the Optical drive has expired. I cannot find the Apple article, but if you change the system date to something older in System Preferences -> Date & Time to say a date in 2012 and then test.

Please also see Error: The installation failed. An unknown installation occurred when installing bundled applications.

Do you have any pending Firmware Updates for your Mac?

Can you install Mavericks on an external disk and boot from it and test you Optical Drive?

You may also want to try an SMC reset and PRAM reset - Startup key combinations for Intel-based Macs - Apple Support.