Horn on a 2010 silverado will not work either from the steering wheel or key fob

You have all the informations to find your problem. jump a wire on connector X4 G1 tan wire to ground and check X1 E5 with a test light or multimeter to see if there is a output, you should have 12 volts. If you dont have any voltage the horn relay is defective. I check in the shop manual and the relay come with the underhood fuse block.

There is also a horn Relay...This sounds like an open ground wire which is beneath the steering wheel plastic cover and or the horn itself which is located around front bumper/frame area or under hood...Use safety glasses with park brake engaged before checking !! ...The ground wire under the steering wheel activates the horn when pressed together which touches the wire to the ground circuit under steering wheel cover...AIR BAG CAN EXPLODE FROM ABUSIVE CONTACT WITH STEERING WHEEL AND WILL HURT YOU BAD !!!..I DO NOT RECOMMEND CHECKING IT ..TAKE TO A GARAGE...If you do try to check under steering cover ALWAYS DISCONNECT BATTERY BEFORE TOUCHING IT...I would replace the Relay without trying to figure out how to test it just to be sure..,check power and ground at horn component before touching steering wheel with an assistant to press horn while checking horn component..,,however in many cases I have found problem to be at steering wheel cover ground...The wire can become loose or disconnected in most situations...DO NOT ATTEMPT STEERING WHEEL COVER REMOVAL IF YOU DONT KNOW WHAT YOU ARE DOING OR WITHOUT DISCONNECTING BATTERY..

The horn is powered from the muilty funtion switch that controls the blinker lights hazared high beams .
IT could be a connection problem at the horn it self but if you are haveing problems these other things small problems anything it could be sign of a different connection problem in the main connection at the harnest going to the steering wheel area look at the muilty funtion switch

You either have a wiring problem or a failed bcm.

I am going with this because you say the horns work when power supplied to fuse - that says the horns are working from the underhood fuse block to horns. Also you stated that it doesn't work with key fob or horn button - which leaves be to believe a wiring problem or bcm failure.