The Apple Digital AV Adapter (HDMI support for iPad, iPhone and iPod touch) will work on the iPad 1 to output video and audio from apps like Videos, Youtube, Keynote etc... but will not work for full mirroring.
Full mirroring is only available on an iPad2 with a Digital AV Adapter. The outputting of both the audio and video through the HDMI cable is good though, one less cable required. Another advantage is that with the second built-in 30-pin connector on the Digital AV Adaptor both the iPad and iPad 2 can be charged while it’s mirroring. The Digital AV adaptor is available at most stores that stock apple accessories for about $A45.
Using the Digital AV adapter
If content does not scale correctly (for example, if content appears stretched or squished), check the Widescreen setting in Settings > iPod or in Settings > Video and the settings of your TV or display.
Note: You may need to toggle the Widescreen setting off and on after connecting if it was initially set to Off when you connected the device to the display.
For optimal performance, you may need to adjust the video resolution or settings for your display. If your display offers an "auto detect" or "factory default" setting, you may be able to use these options to optimize video resolution and display. (from apple support).
Watch this Youtube video which explains how the Digital Av adaptor works with the iPad, iPad2, iPhone and iPod touch.