Everything seems to be working except the starter. No clicking or any other noises from it.
It wont start. Battery is good, lights work,horn works. on 2001 GMC Sierra 2500 HD
If the " security" lamp is not on in the dash cluster it could be the ignition switch, starter motor, or still weak battery or poor cable connection at battery. Next step a technican would do is check for a voltage signal on smaller wire at starter solinoid when ignition is in the crank position.
You may need a starter, if you can take it off,take it to an auto parts store,ask them to test it, they will be able to do this in front of you and show you if it's good or not.