Ok, out of the box, then something is wrong. Thought perhaps if it was 2nd hand, someone botched something else but sold it as "stock".
To check your version number, go to settings and at the bottom, there should be a menu called "about tablet", click on it and there should be one that says "Android version". If it is 4.1.1, you can try to upgrade to the 4.1.2; the update should be pushed out automatically but there are still many that have no received it. Believe there are some threads on here on how to force update. Updating may fix the issue.
Otherwise if you do a factory reset, it will wipe everything and bring it back to the way when you first turned on the device. Play store will save all your apps and you can either download manually or tell it to restore. The option will be there when you first log in. It however does not set up your tablet, so any icons, widgets, etc that you placed on your home screens, you will have to do so again. Also none of your app data will be saved. Such as if you play games, you will lose those save files.
In order to do a factory reset, go to settings > Backup & reset > Factory data reset. Most people do this as a last resort but normally will fix any issues and also make sure you have ample battery when you do it.
And yeah, just search "keyboard" without quotes, in the play store.
As a very last resort, you might want to think about exchanging the unit, but at this point it sounds like a software issue that needs to be corrected, unless for some reason user error but at this point it's difficult to tell which without physically being there. If you do exchange, make sure to do a factory reset anyways.