The only way I've found to make this work, is to have both devices turned off (TV and Nexus10), then connect the HDMI cable, then turn on both the N10 and TV. Once the N10 is half way through the boot sequence, the TV should display everything that's on the Tablet onto the TV.
It's a pain in the ass to have to turn off the tablet and turn it back on... but that's the only solution I've found so far. I read that if you try to connect the mini HDMI while the tablet is on... it could damage it?? I dunno if that's true or not.
<--- Mine. Take a gander if you want.
Free HD Nexus 10 Backgrounds
| Landscape photos taken by me.