Some of the more recent Windows updates have seen issues with installations failing on machines that have removed Microsoft Edge. Fortunately, if you run into this issue there's an easy fix.
Navigate into your 'Program files (x86)' folder on your C: drive and locate the 'Microsoft' folder within. Open this up and delete any folders relating to Microsoft Edge.
Now you can try and install your Windows update again and it should complete without issue. If you follow the steps above, too, Microsoft Edge won't be reinstalled with it.