iPad 2 Broken Screen!
Bummer to hear!

Fortunately, Apple does a great job in allowing you transfer data between devices.

I'm not certain on your meaning of "backup of the files in specific apps"; however, I have two suggestions that may be of help.


Plug your iPad into a trusted computer. This refers to a computer which has been used to sync your iPad in the past. From here you can back up your device with iTunes
. 
[PAID SOLUTION] Third party software is available which will allow you to backup a bit more data than iTunes allows, which is especially useful if you do not plan to purchase another iOS device. I've used a software like iSkySoft iTunes Recovery, which is expensive but does the job well. Search around, I'm sure there are cheapest (or free) solutions if you plan on going the third-party route.


Hope this is of some help to you.