You can purchase a replacement part for your Xperia Z1 through the links further down the page. Additional adhesives for this repair can be found on this page.
This repair isn't particularly difficult but like most does require patience and attention to detail. When we repaired ours there wasn't any pre-cut adhesive available yet so we had to cut our own. You should definitely use the pre-cut type shown lower on the page as it will make your life much easier. It will also help to somewhat maintain the phones IP rating although it is likely that your phone will not be as water resistant once you break it. That said, as long as you don't have any physical damage to the mid-frame or other parts of the phone you should retain a decent amount of water resistance.
Xperia Z1 Compact replacement display
Replacement display for the Z1S
The Z1 was said to have a shatterproof screen protector. We aren't sure if that means it just won't break into a hundred pieces or what. It certainly doesn't mean that this phone can't experience a cracked screen like any other one might. At it's time of production this phone was thought of as one of the best Androids on the market. It scored highly in the ease of use, camera, and battery lifetime categories which is quite an accomplishment. To this day it seems difficult to find a cell phone with enough battery life period.
Interestingly after the release of the Z2 Sony waited only sixth months to release it successor the Z3 which rates well in all categories. Unfortunately Sony is having a hard time gaining market share in the wireless market. Hopefully we will see something exciting from them soon.