Did you transfer the original Home button from your cracked screen? Only that button will allow TouchID to work on your device (they are matched and paired).
If you did transfer the original Home button, then there are three areas you need to inspect with either a magnifying glass or jeweler's loupe. Follow this guide.
1. Where the home button connects to the logic board along with the LCD, Digitizer and Camera.
2. Underneath the shield plate under the Home button, the Home button connects with the Home button extension flex that runs under the LCD heat shield.
3. The tiny circuitry on the Home button flex itself as well as the integrity of the flex between the button and the circuitry.
The connectors at either end of a flex, as well as on the logic board, are very delicate and can be damaged very easily. Check for broken pins or debris that may impede contact. Also, the flex on the Home button tears very easily so look for tears or deep scratches. Finally, because you have to peel off the button flex, look for damage in the area where the electronics components reside.
Always disconnect the battery. It should be the first and last step whenever you open your phone.