With a broken screen you cannot back up data. You cannot prepare the phone for repair by wiping it either
You can have it repaired without wiping the data but the chances are that after screen is replaced the passcode will not work and you will need to wipe the phone with the recovery process anyway.
Good luck, and remember to back up in the future.
Resources
iPhone Service & Repair - Apple Support
Or
Contact - Official Apple Support
Anything other than Screen or Battery may be a replacement phone and the price is in the article under "Other damage"
See also: Get your iPhone, iPad, or iPod touch ready for service - Apple Support
and
How to remove your iPhone, iPad, or iPod touch passcode without a computer - Apple Support
video
https://youtu.be/7s8-zLG9ySQ
ios 15.1 and earlier:
If you've forgotten your iPhone passcode – Apple Support (UK)