Help! I can not find the connector for the home button iPad4

The problem is that you are trying to install a replacement digitizer that is supposed to be for the iPad 4 instead of the iPad 3 one which you should be using.

Basically what you’ll need to do is remove the two screws under the home button on the old screen. Remove the little home button (HB) board from that.

Then remove the two screws at that same spot on the replacement digitizer which has the HB with the cable coming out of it.

Take the board out of that one and use the HB board from the old screen then put the two screws in. Now the HB will work.

TL:DR Can’t use Home Button on a replacement iPad 4 digitizer (as the HB has a cable) on a iPad 3 which you have, need to transfer the HB board from old digitizer.