Here is my solution that doesn't use an Apple TV.
Turn your laptop into an Airplay receiver with this software (Mac & PC compatible: http://www.airserver.com/ ). No HDMI needed. No Apple TV needed. If your Mac or (Windows) PC can connect to a display of any kind (monitor, projector, Smartboard, etc.), you can use it to mirror your iPad's screen.
A few critiques about iPad and Smartboard integration:
First, know that the Smartboard app for iOS is underpowered and disappointing. You cannot control a Smartboard with an iPad. But you can set up your iPad in such a way that you can have nearly all of the Smartboard features. There are free shared whiteboard apps all over the app store. I suggest to get a copy of Keynote and iAnnotate... this will cover 90% of the features of a Smartboard.
Next, know that you cannot operate your iPad from the Smartboard. You can only use the Smartboard to display the screen of your iPad. In terms of classroom modality, this may be desirable. It effectively removes you from the front of the classroom and puts you on the side. You can do nearly everything using an iPad that you can do with the Smartboard... and then so much more.
Finally, you cannot record a class with the iPad like you can with the Smartboard. You may be able to use screen recording software for the Mac or PC running Airplay Server, but I have not tried this and doubt it would work.