I {and a friend of mine} had the same problem and I would like to share my story. Bad home buttons apparently are a common problem with the iPad, but there is a viable, quick, easy and free fix. When I went to the Apple store with my two year old out of warranty iPad they offered to replace it with the same model for $300.00 and take my old iPad. I checked with eBay and noted I could buy the same new one online for $308, including shipping. Plus, I could probably sell my iPad with the buggy home button for more than $100. Some deal.
Then I went to another Apple store and one of the geniuses solved my problem. Go into Settings - General - Accessibility - and click on Assistive Touch. Your new floating home button will appear. You will never need the regular home button again, And this works for iPhones as well.
Best to all.