OK, I'm somewhat embarrassed, but it looks like the problem fixed itself after a couple of days. After noticing the issue first, it was reproducible for a couple of days. I backed up the iPad to iTunes, did a factory reset as a new device, and the issue was still there.
After that, I called Apple Support and they suggested replacing the device, since it seemed to be a hardware issue. I couldn't replace the iPad at this time since I was travelling, so I told them I would wait another couple of days.
When I was back from traveling, I backed up the device again and called Apple Support to arrange the pickup of the iPad. While on the phone with them, I had to describe the issue again and while doing so, was trying to replicate the issue while talking on the phone. Guess what, it worked this time and has worked now for the last couple of days without any issues.
I'm not sure how the issue fixed itself, but there are several possible explanations - not sure how reasonable they are, though:
- The iPad was with me on two atlantic flights (Europe - US and back), maybe the pressure change has fixed the issue, if it was a hardware issue. I remember that the issue was still there when I got to the US, but was gone after I got back.
- Doing another backup/sync with iTunes fixed the issue, which would mean that it was a software issue.
There may be other explanations that I don't see at the moment. While it all is a bit mysterious, I'm glad that it's working now. Let's just hope that it doesn't return.
Update July 2013
The problem returned after a couple of weeks. I finally called Apple Support and they exchanged the device for a new one, since mine was still under warranty. The new one works perfectly fine. So it looks like it was a hardware problem after all.
If you have AssistiveTouch enabled
You can use all the physical features on your device (volume control, on/off button, etc.)
Tap on the dot that appears on your screen. On the popup, tap "Device". Press and hold "Lock Screen", then slide to power off.
Depending on what is causing your device's power button not to work, you may or may not be able to turn it on now in the traditional way by pressing and holding the power button. If that fails, connect your device using its USB cord either to a computer or to a wall socket, and it should turn on after several seconds.
Once the iPad turns on you can enable AssistiveTouch by going to Settings > General > Accessibility > AssistiveTouch (under "Interaction"), and swipe the switch to turn on AssistiveTouch.
AssistiveTouch is an accessibility feature designed for people with limited movement capacities. For more information about AssistiveTouch, see this Apple support article.
If AssistiveTouch is not enabled
You will just have to wait for the battery charge to run out.
Best Answer
Spotlight is the search function in iOS.
It can be accessed from the Home screen by swiping down.
From Apple Support: