There's no help to be had. You broke your phone, and there's no way to back it up if you cannot enter the passcode. There's no workaround.
When you don't back up your data for 6 months, this is what happens. Unfortunate, but hopefully you've learned a valuable lesson.
BTW, are you backing up the data on your computer as well? Hard drives WILL fail. Just a matter of time.....
You might try data recovery services, and I'm truely surprised that Apple would recommend them. A 3rd party replacement of the screen could very well put your device into recovery mode, and if it does, then it has wiped the device and you will not be able to recover. There is no guarantee with data recovery services either, and understand they are expensive. Good luck.