Go to setting on the iPad, check the wifi button is set on. If not, and it is permanently greyed out, reset the settings on your iPad (back it up first) if it still doesn't respond take it into the apple store.
I'd try it in a wi-fi hotspot like MaccyD's, see whether it's something about your router it's objecting to. How does your router identify the devices allowed to connect to it? Is it just a question of entering a password? How many other devices are connected to it?
There's a very long thread about it on apple.com but you could try the advice in one of the last posts:
when i try to connect with simple menu ''choose a network'' , my ipad response ''unable to join..'' ok, just click ON in''ask to join networks'' menu, and go to safari browser , try to connect to a web page and then safari will ask you to connect (different menu to join a network) , so just select your network and insert your password and magically you are in.
It sounds like it isn't doing this - you go to Google but is that in Safari or in the Google app?
And at the risk of committing Apple heresy, have you tried powering the iPad off and back on again?