Hi there -
That's an odd situation. How many APs do you have in the house? Since the ICX is the main power source for them, having a UPS on it would be your best bet.
When you look at your Unleashed dashboard, are all of your APs still up and in the mesh, or are any offline?
As stated above, the R510 does take about 5-10 minutes to fully boot, but I can't imagine you'd have issues once the mesh is back up and fully functional.
There are a few things you can try power wise - do you have access to the CLI of the ICX?