I can only suggest the following:
1. Try to fix yourself by disassembling and putting in a new screen. (voids warranty)
2. Go to a non-branded service outlet and ask them to replace the screen. (may or may not void warranty)
3. Convince the person operating the samsung service center to not perform a wipe. (I know for a fact that a wipe is not necessary to change the screen, but it's a protocol that the employees are asked to follow)
More importantly if you had your sync setup your contacts and agenda is all safe and sound on your google account. You can verify this by visiting your google account on a PC.
Your photos and videos are safe on the internal SD card, normally a wipe or even factory reset does not delete the SD card contents and even following a wipe the data should be safely there. Only a hard wipe (doing factory reset via code) will delete your internal SD card data.
If the samsung center does not do a hard wipe and only does a soft wipe your photos and videos will be safe, you can request them to not perform a hard reset.
the only thing that you shall lose are your messages.
Since the screen is ruined and it is not responding to touch there is only one way that you can turn usb debugging mode on/off.
It will take you an identical S2 on the same os (gingerbread or ICS), and 2 identical usb mice, and 2 usb OTG cables. And a whole lot of precision.
connect the 2 usb otg cables and then connect the usb mice to them. a pointer shall pop up on the working s2. Start replicating the actions of the working S2 on the broken S2. With some luck and a whole lot of precision you can toggle the usb debugging option.
Make sure that before you begin replicating actions, that both pointers are similarly positioned. to do that I advise moving the mouse in such a way that the pointer starts off from a corner of the screen.
For future use one of the many free sms backup options available on Play store. Some like SMS Backup+ auto backup your messages to your gmail account.
For your Photos and Videos I recommend using dropbox. It has a feature by which all your clicked photos and videos get saved on the dropbox account also and via cloud syncing also get saved to your PC. You can alternatively use the Google+ instant upload feature where all your photos get saved on a private album on your G+ account.
Hope this helps