Yup label your gear, near the inflate valve on the LE as suggested sounds like a sensible spot, and make it big enough that someone that has never seen a kite had half a chance of finding and reading it.
There is no hard and fast rule as to when a marine search is kicked off and I can say from first hand experience they are not started on a whim because a bunch of blokes are bored and want to put training into practice. The guys at water police, air wing, VMR's and the RAC helicopter get plenty of searches and false alarms without kicking off any extras. Making the decision to start a marine search sucks and I'd be happy to never have to do it again.

Water police occassionally get calls from people sitting in cafes and on the beach after they see a kite go down out to sea and not relaunch. Water Police know we self-rescue, boards are easily lost and we almost always only go out in onshore winds, all these work in our favour so they won't necessarily start a search immediately. A limp kite washed up on the beach without someone attached to it (or chasing it up the beach) is unusual and would be taken more seriously and I suspect that is what happened at City Beach a while back.
If in doubt call Water Police, 9442 8600 (WA) - 24 hours. A search will ALWAYS be started if necessary but it is nice to call them off before it gets to that point.