I have found the service manual for the Kd 55x9000a, thinking there maybe a answer in it to your problem. This tv should be similar to yours. http://portaldaeletronica.com/wp-content/uploads/2016/02/Manual-de-servi%C3%A7o-TV-SONY-KD-55X9000A-chassis-RB1FS.pdf
If this is a landline, ensure that your service is working. If you do have service then check the line between where the phone comes into the house, and the phone base. It sounds like the phone is charged, cause you are getting a searching for service notice. If nothing is wrong with the service or wire, then it has to be the phone itself. Here is the link to reset the base. https://unidenamerica.desk.com/customer/...-perform-factory-reset-on-cordless-handset?b_id=16266
If you have ruled out the charger then it will probably either be the battery, or the connection in between the battery and the plug charger. Did you make sure everything was hooked back up after you replaced your screen. The battery sounds like it was still attached since you played on the phone for 5 minutes before the power went out. So check at the point where you plug the charger and how it attaches to the battery.
You can superglue it, only thing is it will be hard to get it apart if you need to fix the screen again. It sounds like there may be either some residue from the other times you put it in or it just isn’t seated correctly. If worse comes to worse, take it back apart, check for anything that may keep it from seating correctly, (clean off any excess of anything), and check to ensure that nothing is in the way, (something could have shifted or isn’t attached where it originally was), then try putting it together again. (If the pads you are referring to have any type of wires coming from them, then they are probably needed to make the device work.